nonlinear partial differential equations with applications ||

ISNMInternational Series of Numerical Mathematics

Volume 153

Managing Editors:K.-H. Hoffmann, München, GermanyG. Leugering, Erlangen-Nürnberg, Germany

Associate Editors:Z. Chen, Beijing, ChinaR.H.W. Hoppe, Augsburg, Germany/Houston, USAN. Kenmochi, Chiba, JapanV. Starovoitov, Novosibirsk, Russia

Honorary Editor:J. Todd, Pasadena, USA†

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Nonlinear Partial DifferentialEquations with Applications

Tomáš Roubí ek

Second Edition

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05, 201

ISBN 978-3-0348-0512- ISBN 978-3-0348-0513-1 (eBook) DOI 10.1007/978-3-0348-0513-1 Springer Basel Heidelberg New York Dordrecht London

4

Mathematics Subject Classification (2010): Primary: 35Jxx, 35Kxx, 35Qxx, 47Hxx, 47Jxx, 49Jxx: , x, xx, 7 xx, xxSecondary 65Nxx 74Bx 74F 6D 80A

Tomáš Roubíček Mathematical Institute Charles University Sokolovská 83 186 75 Praha 8 Czech Republic

Institute of Thermomechanics of the ASCR Dolejškova 5 182 00 Praha 8 Czech Republic

Institute of Information Theory and Automation of the ASCR Pod vodárenskou věží 4 182 08 Praha 8 Czech Republic

and

and

;

Library of Congress Control Number: 20 21 956219

http://www.birkhauser-science.com

To the memory of professor Jindrich Necas

Contents

Preface xi

Preface to the 2nd edition xv

Notational conventions xvii

1 Preliminary general material 11.1 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Normed spaces, Banach spaces, locally convex spaces . . . . 11.1.2 Functions and mappings on Banach spaces, dual spaces . . 31.1.3 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.5 Fixed-point theorems . . . . . . . . . . . . . . . . . . . . . 8

1.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Continuous and smooth functions . . . . . . . . . . . . . . . 91.2.2 Lebesgue integrable functions . . . . . . . . . . . . . . . . . 101.2.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Nemytskiı mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 Green formula and some inequalities . . . . . . . . . . . . . . . . . 201.5 Bochner spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6 Some ordinary differential equations . . . . . . . . . . . . . . . . . 25

I STEADY-STATE PROBLEMS 29

2 Pseudomonotone or weakly continuous mappings 312.1 Abstract theory, basic definitions, Galerkin method . . . . . . . . . 312.2 Some facts about pseudomonotone mappings . . . . . . . . . . . . 352.3 Equations with monotone mappings . . . . . . . . . . . . . . . . . 372.4 Quasilinear elliptic equations . . . . . . . . . . . . . . . . . . . . . 42

2.4.1 Boundary-value problems for 2nd-order equations . . . . . . 432.4.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . 442.4.3 Pseudomonotonicity, coercivity, existence of solutions . . . 48

vii

viii Contents

2.4.4 Higher-order equations . . . . . . . . . . . . . . . . . . . . . 562.5 Weakly continuous mappings, semilinear equations . . . . . . . . . 61

2.6 Examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . 64

2.6.1 General tools . . . . . . . . . . . . . . . . . . . . . . . . . . 642.6.2 Semilinear heat equation of type −div(A(x, u)∇u) = g . . . 68

2.6.3 Quasilinear equations of type −div(|∇u|p−2∇u

)+c(u,∇u)=g 75

2.7 Excursion to regularity for semilinear equations . . . . . . . . . . . 85

2.8 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 92

3 Accretive mappings 95

3.1 Abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.2 Applications to boundary-value problems . . . . . . . . . . . . . . 99

3.2.1 Duality mappings in Lebesgue and Sobolev spaces . . . . . 99

3.2.2 Accretivity of monotone quasilinear mappings . . . . . . . . 101

3.2.3 Accretivity of heat equation . . . . . . . . . . . . . . . . . . 1053.2.4 Accretivity of some other boundary-value problems . . . . . 108

3.2.5 Excursion to equations with measures in right-hand sides . 109

3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.4 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 114

4 Potential problems: smooth case 115

4.1 Abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2 Application to boundary-value problems . . . . . . . . . . . . . . . 120

4.3 Examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . 126


5 Nonsmooth problems; variational inequalities 133

5.1 Abstract inclusions with a potential . . . . . . . . . . . . . . . . . 133

5.2 Application to elliptic variational inequalities . . . . . . . . . . . . 1375.3 Some abstract non-potential inclusions . . . . . . . . . . . . . . . . 145

5.4 Excursion to quasivariational inequalities . . . . . . . . . . . . . . 154

5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.6 Some applications to free-boundary problems . . . . . . . . . . . . 163

5.6.1 Porous media flow: a potential variational inequality . . . . 163

5.6.2 Continuous casting: a non-potential variational inequality . 166


6 Systems of equations: particular examples 171

6.1 Minimization-type variational method: polyconvex functionals . . . 1716.2 Buoyancy-driven viscous flow . . . . . . . . . . . . . . . . . . . . . 178

6.3 Reaction-diffusion system . . . . . . . . . . . . . . . . . . . . . . . 186

6.4 Thermistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.5 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Contents ix

II EVOLUTION PROBLEMS 199

7 Special auxiliary tools 2017.1 Sobolev-Bochner space W 1,p,q(I;V1, V2) . . . . . . . . . . . . . . . 2017.2 Gelfand triple, embedding W 1,p,p′

(I;V ,V ∗)⊂C(I;H) . . . . . . . . 2047.3 Aubin-Lions lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8 Evolution by pseudomonotone or weakly continuous mappings 2138.1 Abstract initial-value problems . . . . . . . . . . . . . . . . . . . . 2138.2 Rothe method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2158.3 Further estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2308.4 Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2408.5 Uniqueness and continuous dependence on data . . . . . . . . . . . 2478.6 Application to quasilinear parabolic equations . . . . . . . . . . . . 2518.7 Application to semilinear parabolic equations . . . . . . . . . . . . 2618.8 Examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . 264

8.8.1 General tools . . . . . . . . . . . . . . . . . . . . . . . . . . 2648.8.2 Parabolic equation of type ∂

∂tu−div(|∇u|p−2∇u)+c(u)=g . 266

8.8.3 Semilinear heat equation c(u) ∂∂tu− div(κ(u)∇u) = g . . . . 277

8.8.4 Navier-Stokes equation ∂∂tu+(u·∇)u−Δu+∇π=g, div u=0 . 279

8.8.5 Some more exercises . . . . . . . . . . . . . . . . . . . . . . 2828.9 Global monotonicity approach, periodic problems . . . . . . . . . . 2888.10 Problems with a convex potential: direct method . . . . . . . . . . 2948.11 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 300

9 Evolution governed by accretive mappings 3039.1 Strong solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3039.2 Integral solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3089.3 Excursion to nonlinear semigroups . . . . . . . . . . . . . . . . . . 3149.4 Applications to initial-boundary-value problems . . . . . . . . . . . 3199.5 Applications to some systems . . . . . . . . . . . . . . . . . . . . . 3269.6 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 332

10 Evolution governed by certain set-valued mappings 33510.1 Abstract problems: strong solutions . . . . . . . . . . . . . . . . . . 33510.2 Abstract problems: weak solutions . . . . . . . . . . . . . . . . . . 33910.3 Examples of unilateral parabolic problems . . . . . . . . . . . . . . 34310.4 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 349

11 Doubly-nonlinear problems 35111.1 Inclusions of the type ∂Ψ( d

dtu) + ∂Φ(u) � f . . . . . . . . . . . . . 35111.1.1 Potential Ψ valued in R ∪ {+∞}. . . . . . . . . . . . . . . . 35111.1.2 Potential Φ valued in R ∪ {+∞} . . . . . . . . . . . . . . . 35811.1.3 Uniqueness and continuous dependence on data . . . . . . . 366

x Contents

11.2 Inclusions of the type ddtE(u) + ∂Φ(u) � f . . . . . . . . . . . . . . 367

11.2.1 The case E := ∂Ψ. . . . . . . . . . . . . . . . . . . . . . . . 36811.2.2 The case E non-potential . . . . . . . . . . . . . . . . . . . 37211.2.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

11.3 2nd-order equations . . . . . . . . . . . . . . . . . . . . . . . . . . 37711.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38511.5 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . . 390

12 Systems of equations: particular examples 39312.1 Thermo-visco-elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 39312.2 Buoyancy-driven viscous flow . . . . . . . . . . . . . . . . . . . . . 40512.3 Predator-prey system . . . . . . . . . . . . . . . . . . . . . . . . . 40812.4 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41212.5 Phase-field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41612.6 Navier-Stokes-Nernst-Planck-Poisson-type system . . . . . . . . . . 42012.7 Thermistor with eddy currents . . . . . . . . . . . . . . . . . . . . 42612.8 Thermodynamics of magnetic materials . . . . . . . . . . . . . . . 43212.9 Thermo-visco-elasticity: fully nonlinear theory . . . . . . . . . . . . 438

449

Index 469

Bibliography

Preface

The theoretical foundations of differential equations have been significantly devel-oped, especially during the 20th century. This growth can be attributed to fast andsuccessful development of supporting mathematical disciplines (such as functionalanalysis, measure theory, and function spaces) as well as to an ever-growing callfor applications especially in engineering, science, and medicine, and ever betterpossibility to solve more and more complicated problems on computers due toconstantly growing hardware efficiency as well as development of more efficientnumerical algorithms.

A great number of applications involve distributed-parameter systems (whichcan be, in particular, described by partial differential equations1) often involvingvarious nonlinearities. This book focuses on the theory of such equations with theaim of bringing it as fast as possible to a stage applicable to real-world tasks. Thiscompetition between rigor and applicability naturally needs many compromises tokeep the scope reasonable. As a result (or, conversely, the reason for it) the book isprimarily meant for graduate or PhD students in programs such as mathematicalmodelling or applied mathematics. Although some preliminary knowledge of mod-ern methods in linear partial differential equations is useful, the book is basicallyself-contained if the reader consults Chapter 1 where auxiliary material is brieflypresented without proofs.

The prototype tasks addressed in this book are boundary-value problems forsemilinear2 equations of the type

−Δu+ c(u) = g, or more general − div(κ(u)∇u

)+ c(u) = g, (0.1)

or, still more general, for quasilinear3 equations of the type

−div(a(u,∇u)

)+ c(u,∇u) = g, (0.2)

1The adjective “partial” refers to occurrence of partial derivatives.2In this book the adjective “semilinear” will refer to equations where the highest derivatives

stand linearly and the induced mappings on function spaces are weakly continuous.3The adjective “quasilinear” refers to equations where the highest derivatives occur lin-

early but multiplied by functions containing lower-order derivatives, which means here the form−∑n

i,j=1 aij(x, u,∇u)∂2u/∂xi∂xj + c(x, u,∇u) = g. After applying the chain rule, one can see

that (0.2) is only a special case, namely an equation in the so-called divergence form.

ix

xii Preface

and various generalizations of those equations, in particular variational inequali-ties. Furthermore, systems of such equations are treated with emphasis on variousreal-world applications in (thermo)mechanics of solids and fluids, in electrical de-vices, engineering, chemistry, biology, etc. These applications are contained inPart I.

Part II addresses evolution variants of previously treated boundary-valueproblems like, in case of (0.2),4

∂u

∂t− div

(a(u,∇u)

)+ c(u,∇u) = g, (0.3)

completed naturally by boundary conditions and initial or periodic conditions.Let us emphasize that our restriction on the quasilinear equations (or in-

equalities) in the divergence form is not severe from the viewpoint of applica-tions. However, in addition to fully nonlinear equations of the type a(Δu) = g or∂∂tu+a(Δu) = g, topics like problems on unbounded domains, homogenization, de-tailed qualitative aspects (asymptotic behaviour, attractors, blow-up, multiplicityof solutions, bifurcations, etc.) and, except for a few remarks, hyperbolic equationsare omitted.

In particular cases, we aim primarily at formulation of a suitable definitionof a solution and methods to prove existence, uniqueness, stability or regularity ofthe solution.5 Hence, the book balances the presentation of general methods andconcrete problems. This dichotomy results in two levels of discourse interactingwith each other throughout the book:

• abstract approach – can be explained systematically and lucidly, has its owninterest and beauty, but has only an auxiliary (and not always optimal)character from the viewpoint of partial differential equations themselves,

• targeted concrete partial differential equations – usually requires many tech-nicalities, finely fitted with particular situations and often not lucid.

The addressed methods of general purpose can be sorted as follows:

◦ indirect in a broader sense: construction of auxiliary approximate problemseasier to solve (e.g. Rothe method, Galerkin method, penalization, regular-ization), then a-priori estimates and a limit passage;

◦ direct in a broader sense: reformulation of the differential equation or inequal-ity into a problem solvable directly by usage of abstract theoretical results,e.g. potential problems, minimization by compactness arguments;

◦ iterational: fixed points, e.g. Banach or Schauder’s theorems;

4In fact, a nonlinear term of the type c(u) ∂∂t

u can easily be considered in (0.3) instead of ∂∂t

u;see p. 277 for a transformation to (0.3) or Sect. 11.2 for a direct treatment. Besides, nonlinearitylike C( ∂

∂tu) will be considered, too; cf. Sect. 11.1.1 or 11.1.2.

5To complete the usual mathematical-modelling procedure, this scheme should be precededby a formulation of the model, and followed by numerical approximations, numerical analysis,with computer implementation and graphic visualization. Such, much broader ambitions are notaddressed in this book, however.

Preface xiii

We make the general observation that simple problems usually allow several ap-proaches while more difficult problems require sophisticated combination of manymethods, and some problems remain even unsolved.

The material in this book is organized in such a way that some material canbe skipped without losing consistency. At this point, Table 1 can give a hint:

steady-state evolution

Chapter 7,basic minimal scenario Chapters 2,4Sect. 8.1–8.8

variational inequalities Chapter 5 Chapter 10

accretive setting Chapter 3 Chapter 9

Chapter 12,systems of equations Chapters 6Sect. 9.5

Sect. 8.9–8.10,some special topics —Chapter 11

auxiliary summaryof general tools Chapter 1

Table 1. General organization of this book.

Except for the basic minimal scenario, the rest can be combined (or omitted)quite arbitrarily, assuming that the evolution topics will be accompanied by thecorresponding steady-state part. Most chapters are equipped with exercises whosesolution is mostly sketched in footnotes. Suggestions for further reading as well assome historical comments are in biographical notes at the ends of the chapters.

The book reflects both my experience with graduate classes I taught in theprogram “Mathematical modelling” at Charles University in Prague during 1996–20056 and my own research7 and computational activity in this area during thepast (nearly three) decades, as well as my electrical-engineering background andresearch contacts with physicists and material scientists. My thanks and deep

6In the usual European 2-term organization of an academic year, a natural schedule was Part I(steady-state problems) for one term and Part II (evolution problems) for the other term. Yet,only a selection of about 60% of the material was possible to expose (and partly accompaniedby exercises) during a 3-hour load per week for graduate- or PhD-level students. Occasionally, Ialso organized one-term special “accretive-method” course based on Chapters 3 and 9 only.

7It concerns in particular a research under the grants 201/03/0934 (GA CR), IAA 1075402(GA AV CR), and MSM 21620839 (MSMT CR) whose support is acknowledged.

xiv Preface

gratitude are to a lot of my colleagues, collaborators, or tutors, in particularM. Arndt, M. Benes, M. Bulıcek, M. Feistauer, J. Francu, J. Haslinger, K.-H. Hoff-mann, J. Jarusek, J. Kacur, M. Kruzık, J. Malek, J. Maly, A. Mielke, J. Necas,M. Pokorny, D. Prazak, A. Swierczewska, and J. Zeman for numerous discussionsor/and reading the manuscript.

Praha, 2005 T.R.

Preface to the 2nd edition

Although the core of the book is identical with the 1st edition, at particular spotsthis new edition modifies and expands it quite considerably, reflecting partly thefurther research of my own8 and my colleagues, as well as a feedback from contin-uation of classes in the program “Mathematical modelling” at Charles Universityin Prague during 2005–2012, partly executed also by my colleague, M.Bulıcek.

More specifically, the main changes are as follows: On an abstract level,Rothe’s method has been improved by using a finer discrete Gronwall inequalityand by refining some estimates to work if the governing potential is only semicon-vex, as well as various semi-implicit variants have been added. Also the presen-tation of Galerkin’s method in Sect. 8.4 has been simplified. Morever, needless tolist, some particular assertions have been strengthened or their proofs simplified.

On the level of concrete partial differential equations, boundary conditionsin higher-order equations in Section 2.4.4 are now more elaborated. Interpolationby using Gagliardo-Nirenberg inequality has been applied more systematically; inparticular the exponent p� has been defined more meticuously and a new “bound-ary” exponent p#© has been introduced and used in a modified presentation of theparabolic equations in Sect. 8.6. Interpolation has also been exploited in new esti-mates especially in examples of thermally coupled systems which have been moreelaborated or completely rewritten, cf. Sects. 6.2 and 12.1, and even some newerones have been added, cf. Sects. 12.7–12.9. Other issues concern e.g. newly addedsingularly-perturbed problems, positivity of solutions (typically temperature inthe heat equation), Navier’s boundary conditions, etc.

Moreover, some rather local augments have been implemented. Various exer-cises have been expanded or added and some new applications have been involved.Also the list of references has been expanded accordingly. Of course, various typosor mistakes have been corrected, too.

Last but not least, it should be emphasized that inspiring discussions withM.Bulıcek, J.Malek, J.Maly, P. Podio-Guidugli, U. Stefanelli, and G. Tomassettihave thankfully been reflected in this 2nd edition.

Praha, 2012 T.R.

8In particular it concerns the GA CR-projects 201/09/0917, 201/10/0357, and 201/12/0671.

xv

Notational conventions

A a mapping (=a nonlinear operator), usually V → V ∗ or dom(A) → X ,

a.a., a.e. a.a.=almost all, a.e.=almost each, referring to Lebesgue measure,

C(Ω) the space of continuous functions on Ω, equipped with the norm‖u‖C(Ω) = maxx∈Ω |u(x)|, sometimes also denoted by C0(Ω),

C0,1(Ω) the space of the Lipschitz continuous functions on Ω,

C(Ω;Rn) the space of the continuous Rn-valued functions on Ω,

Ck(Ω) the space of functions whose all derivatives up to k-th order are con-tinuous on Ω,

cl(·) the closure,

curl the rotation of a vector-valued field on R3, see p. 181,

D(Ω) the space of infinitely smooth functions with a compact support in Ω,see p. 10,

DΦ(u, v) the directional derivative of Φ at u in the direction v,

diam(S) the diameter of a set S ⊂ Rn; i.e. diam(S) := supx,y∈S |x− y|,div the divergence of a vector field; i.e. div(v) = ∂

∂x1v1 + · · · + ∂

∂xnvn for

v = (v1, . . . , vn),

divS the surface divergence, divS := Tr(∇S), see p. 57,

dom(A) the definition domain of the mapping A; in case of a set-valued mappingA : V1 ⇒ V2 we put dom(A) := {v∈V1; A(v) �= ∅},

dom(Φ) the domain of Φ : V → R ∪ {+∞}; dom(Φ) := {v∈V ; Φ(v) < +∞},e(·) the symmetric gradient, cf. 22,

epi(Φ) the epigraph of Φ; i.e. {(v, a)∈V ×R; a ≥ Φ(v)},I the time interval [0, T ],

I the identity mapping,

I the unit matrix,

int(·) the interior,

J the duality mapping,

L (V1, V2) the Banach space of linear continuous mappings A : V1 → V2 normedby ‖A‖L (V1,V2) = sup‖v‖V1≤1 ‖Av‖V2 ,

Lp(Ω) the Lebesgue space of p-integrable functions on Ω, equipped with the

norm ‖u‖Lp(Ω) =( ∫

Ω |u(x)|pdx)1/p,Lp(Ω;Rn) the Lebesgue space of Rn-valued p-integrable functions on Ω,

M (Ω) the space of regular Borel measures, M (Ω) ∼= C(Ω)∗, cf. p.10,measn(·) n-dimensional Lebesgue measure of a set,

iixv

xviii Notational conventions

n the spatial dimension,

N the set of all natural numbers,

Na the Nemytskiı mapping induced by an integrand a,

NK(·) the normal cone, cf. p.6,

O(·) the “great O” symbol: f(ε) = O(εα) for ε↘0 means lim supε↘0

|f(ε)|εα

<∞,

o(·) the “small O” symbol: f(ε) = o(εα) for ε↘0 means limε↘0 f(ε)/εα = 0,

p the exponent related to the polynomial growth/coercivity of thehighest-order term in a differential operator,

p′ = pp−1 the conjugate exponent to p ∈ [1,+∞], cf. (1.20) on p.12,

p∗ the exponent in the embedding W 1,p(Ω) ⊂ Lp∗(Ω), see (1.34)on p.16,

p∗∗ the exponent in the embedding W 2,p(Ω) ⊂ Lp∗∗(Ω), i.e. p∗∗ = (p∗)∗,

p# the exponent in the trace operator u �→ u|Γ : W 1,p(Ω) → Lp#

(Γ), see

(1.37) on p.17; e.g. p#′or p∗#′ mean (p#)′ or ((p∗)#)′, respectively,

p� the exponent in the continuous embeddingLp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)) ⊂ Lp�

(Q), see (8.131) on p.253,

p#© the exponent in the trace operator Lp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)) →Lp#©

(Σ), see (8.136) on p.254,

Q a space-and-time cylinder, Q = I × Ω,

R, R+, R− the set of all (resp. positive, or negative) reals,

R the set of extended reals R ∪ {+∞,−∞},Rn the Euclidean space with the norm |s| = |(s1, . . . , sn)| =

(∑ni=1 s

2i

)1/2,

sign the single-valued “signum”, i.e. the mapping R → [−1, 1], sign(0) = 0,sign(R+) = 1, sign(R−) = −1, cf. Figure 10a on p.133,

Sign the set-valued “signum”, i.e. the mapping R ⇒ [−1, 1], Sign(0) =[−1, 1], Sign(R+) = {1}, Sign(R−) = {−1}, cf. Figure 10b on p.133,

span(·) the linear hull of the specified set,

supp(u) the support of a function u, i.e. the closure of {x ∈ Ω; u(x) �= 0},T a fixed time horizon, T > 0,

V a separable reflexive Banach space (if not said otherwise), ‖ · ‖V (orbriefly ‖ · ‖) its norm,

V ∗ a topological dual space with ‖ · ‖V ∗ (or briefly ‖ · ‖∗) its norm,

W k,p(Ω) the Sobolev space of functions whose distributional derivatives up tokth order belongs to Lp(Ω), cf. (1.30) on p.15.

W 1,p0 (Ω) the Sobolev space of functions from W 1,p(Ω) whose traces on Γ vanish,

W−1,p′(Ω) the dual space to W 1,p

0 (Ω),

W k,ploc (Ω) the set of functions v on Ω whose restrictions v|O, with any open O

such that O ⊂ Ω, belong to W k,p(O),

Notational conventions xix

W 1,p,q the Sobolev space of abstract functions having the time-derivative,see (7.1) on p. 201,

W 1,p,M the Sobolev space of abstract functions whose derivatives are measures,see (7.40) on p. 211,

W 2,∞,p,q the Sobolev space of abstract functions having the second time-derivative, see (7.4) on p. 202,

W 1,p0,div the set of divergence-free functions v∈W 1,p

0 (Ω;Rn), see (6.29) on p. 178,

Γ the boundary of a domain Ω,

δK the indicator function of a set K; i.e. δK(·) = 0 on K and δK(·) = +∞on the complement of K,

δx the Dirac distribution (measure) supported at a point x,

Δ the Laplace operator: Δu = div(∇u) = ∂2

∂x21u+ · · ·+ ∂2

∂x2nu,

Δp the p-Laplace operator: Δpu = div(|∇u|p−2∇u) with p > 1,

ν the unit outward normal to Γ at x ∈ Γ, ν = ν(x),

Σ the side surface of the cylinder Q, i.e. I×Γ, or a σ-algebra of sets,

χS the characteristic function of a set S; i.e. χS(·) = 1 on S and χS(·) = 0on the complement of S,

Ω a bounded, connected, Lipschitz domain, Ω ⊂ Rn,

Ω the closure of Ω,

⊂ a subset, or a continuous embedding,

� a compact embedding,∫Ω . . . dx integration according to the n-dimensional Lebesgue measure,∫Γ. . .dS integration according to the (n−1)-dimensional surface measure on Γ,

∂Φ the subdifferential of the convex functional Φ : V → R,(·)−1 the inverse mapping,

(·)∗ the dual space, see p.3, or the adjoint operator, see p.5, or the Legendre-Fenchel conjugate functional, see p.294,

(·)+, (·)− the positive and the negative parts, respectively, i.e. u+ = max(u, 0)and u− = min(u, 0),

(·)′ the Gateaux derivative, cf. p.5, or a partial derivative, or the conjugateexponent, see p.12,

(·)|S the restriction of a mapping or a function on a set S,

(·) the transposition of a matrix,

→ a convergence (in a locally convex space) or a mapping between sets,

�→ a mapping of elements into other ones, e.g. A : u �→ f where f = A(u),

↘ convergence on R from the right; similarly ↗ means from the left,

⇒ a set-valued mapping (e.g. A : X ⇒ Y abbreviates A : X → 2Y = theset of all subsets of Y ),

xx Notational conventions

∇ the spatial gradient: ∇u = ( ∂∂x1

u, . . . , ∂∂xn

u),

∇S the surface gradient, see p. 57,

· a position of an unspecified variable, or the scalar product of vectors;i.e. u·v :=

∑mi=1 uivi for u, v ∈ Rm,

: the scalar product of matrices; i.e. A:B :=∑n

i=1

∑mj=1 AijBij ,.

: the product of 3rd-order tensors A.:B :=

∑ni=1

∑mj=1

∑lk=1 AijkBijk,

:= the definition of a left-hand side by a right-hand-side expression,

⊗ the tensorial product of vectors: [u⊗ v]ij = uivj ,

〈·, ·〉 the bilinear pairing of spaces in duality, cf. p.3,

〈·, ·〉s the semi-inner product in a Banach space, cf. (3.7) on p.97,

(·, ·) the inner (i.e. scalar) product in a Hilbert space, cf. (1.4) on p.2,

‖ · ‖ a norm on a Banach space, see p.1,

| · | a seminorm on a Banach space, or an Euclidean norm in Rn.

Chapter 1

Preliminary general material

For the reader’s convenience, this chapter summarizes some concepts, definitionsand results which are mostly relevant to the undergraduate curriculum and arethus assumed as basically known, or have specific roots in rather distant areasand have rather auxiliary character with respect to the purpose of this book,which is to push pure theory towards possible applications as fast as possible. Assuch, all assertions in Chapter 1 are made without proofs and the scope has beenminimized to only material actually needed in the book. If a reader decides to skipthis Chapter, (s)he can easily come back through references in the Index or in thetext itself, if in need.

1.1 Functional analysis

The universal framework used in (even nonlinear) differential equations is basedon linear functional analysis, and also on convex analysis.

1.1.1 Normed spaces, Banach spaces, locally convex spaces

Considering a (real) linear space V ,1 a non-negative, degree-1 homogeneous, sub-additive functional ‖ · ‖V : V → R is called a norm if it vanishes only at 0; often,we will write briefly ‖ · ‖ instead of ‖ · ‖V if V is clear from the context.2 A linearspace equipped with a norm is called a normed linear space. If the last property(i.e. ‖u‖V = 0 ⇒ u = 0) is missing, we call such a functional a seminorm; i.e. a

1This means V is endowed by a binary operation (v1, v2) �→ v1 + v2 : V × V → V whichmakes it a commutative group, i.e. v1 + v2 = v2 + v1, v1 + (v2 + v3) = (v1 + v2) + v3, ∃ 0∈V :v + 0 = v, and ∀v1 ∈ V ∃v2: v1 + v2 = 0, and furthermore it is equipped with a multiplicationby scalars (a, x) �→ ax : R× V → V satisfying (a1 + a2)v = a1v + a2v, a(v1 + v2) = av1 + av2,(a1a2)v = a1(a2v), and 1v = v.

2The mentioned properties mean respectively: ‖v‖ ≥ 0, ‖av‖ = |a| ‖v‖, ‖u+v‖ ≤ ‖u‖ + ‖v‖for any u, v ∈ V and a ∈ R, and ‖v‖ = 0 ⇒ v = 0.

1T. Roubí ek, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics 153, DOI 10.1007/978-3-0348-0513-1_1, © Springer Basel 2013

2 Chapter 1. Preliminary general material

functional | · |ξ : V → R is a seminorm if it satisfies

∀u, v∈V ∀a∈R : 0 ≤ |u+v|ξ ≤ |u|ξ + |v|ξ and |au|ξ = |a| |u|ξ. (1.1)

Having V equipped with a collection {| · |ξ}ξ∈Ξ of seminorms | · |ξ with an arbitraryindex set Ξ, we call V a locally convex space. A sequence {uk}k∈N in V is thencalled a Cauchy sequence if

∀ξ∈Ξ ∀ε > 0 ∃k0∈N ∀k1, k2 ≥ k0 : |uk1 − uk2 |ξ ≤ ε. (1.2)

Moreover, a sequence {uk}k∈N is called convergent to some u ∈ V if

∀ξ∈Ξ ∀ε > 0 ∃k0∈N ∀k ≥ k0 : |uk − u|ξ ≤ ε. (1.3)

In this case, u is called its limit and we will write u = limk→∞ uk or uk → u (or,depending on a particular collection of seminorms, uk ⇀ u). A subset A of a locallyconvex space V is called closed if any limit of any convergent sequence containedin A is itself in A.3 Moreover, A is called open if V \A is closed. The closure of A,denoted by cl(A), is the smallest closed set B ⊃ A, while int(A) := A \ cl(V \A)is called the interior of A. The concrete collection of seminorms will often bespecified by various adjectives as “strong” or “weak” or “weak*”. If |u|ξ = 0 forall ξ ∈ Ξ implies u = 0, V is called a Hausdorff locally convex space. Then everyconvergent sequence has a uniquely determined limit. The normed linear spaceis a Hausdorff locally convex space with its only seminorm being then just thenorm. A subset A ⊂ V is called bounded if supx∈A ‖x‖ < +∞, and dense (in V ) ifcl(A) = V . If there is a countable dense subset of V , we say that V is separable.If every Cauchy sequence in a normed linear space V converges, we say that thisspace is complete and then V is called a Banach space.4 An example of a Banachspace is Rn endowed by the norm, denoted usually by | · | instead of ‖ · ‖, definedby |s| = (

∑ni=1 s

2i )

1/2; such a Banach space is called an n-dimensional Euclideanspace.

If V is a Banach space such that, for any v ∈ V , V → R : u �→ ‖u + v‖2 −‖u−v‖2 is linear, then V is called a Hilbert space. In this case, we define the innerproduct (also called scalar product) by

(u, v) :=1

4‖u+ v‖2 − 1

4‖u− v‖2. (1.4)

By the assumption, (·, ·):V×V→R is a bilinear form which is obviously symmetric5

and satisfies (u, u) = ‖u‖2. E.g., the Euclidean space Rn is a Hilbert space.

3We will always work with Ξ at most countable and therefore, for simplicity, we define “closed-ness” in terms of convergence of sequences, which would be in general situations called rather“sequential closedness”.

4This fundamental concept has been introduced in [33] in 1922.5This means both u �→ (u, v) and v �→ (u, v) are linear functionals on V and (u, v) = (v, u).

1.1. Functional analysis 3

Let us call a Banach space V strictly convex if the sphere6 in V does notcontain any line segment. The space V is uniformly convex if

∀ε > 0 ∃δ > 0 ∀u, v∈V :‖u‖=‖v‖=1‖u− v‖ ≥ ε

}⇒

∥∥∥12u+

1

2v∥∥∥ ≤ 1− δ. (1.5)

Any uniformly convex Banach space is strictly convex but not vice versa.

1.1.2 Functions and mappings on Banach spaces, dual spaces

Having in mind a specific mode of convergence, a function f : V → R ∪ {±∞} iscalled lower (resp. upper) semicontinuous7 if

∀u∈V, uk → u : f(u) ≤ lim infk→∞

f(uk)(resp. f(u) ≥ lim sup

k→∞f(uk)

). (1.6)

Having two normed linear spaces V1 and V2 and a mapping A : V1 → V2, we saythat A is continuous if it maps convergent sequences in V1 onto convergent onesin V2, and is a linear operator if it satisfies A(a1v1 + a2v2) = a1A(v1) + a2A(v2)for any a1, a2 ∈ R and v1, v2 ∈ V1. Often we will write briefly Av instead of A(v).If V1 = V2, a linear continuous operator A : V1 → V2 is called a projector ifA ◦ A = A. The set of all linear continuous operators V1 → V2 will be denotedby L (V1, V2), being itself a normed linear space when equipped with the additionand multiplication by scalars defined respectively by (A1+A2)v = A1v+A2v and(aA)v = a(Av), and with the norm

‖A‖L (V1,V2) := sup‖v‖V1≤1

‖Av‖V2 = supv �=0

‖Av‖V2

‖v‖V1

. (1.7)

A continuous (possibly even nonlinear) mapping A : V1 → V2 is called a homeo-morphism if the inverse A−1 : V2 → V1 does exist and is continuous. Moreover,A : V1 → V2 is called a homeomorphical embedding if A : V1 → A(V1) is a homeo-morphism and A(V1) is dense in V2.

As R itself is a linear topological space8, we can consider the linear spaceL (V,R), being also denoted by V ∗ and called the dual space to V . The originalspace V is then called predual to V ∗. For an operator (=now a functional) f ∈ V ∗,we will write 〈f, v〉 instead of fv. The bilinear form 〈·, ·〉V ∗×V : V ∗ × V → R iscalled a canonical duality pairing. Instead of 〈·, ·〉V ∗×V , we will occasionally writebriefly 〈·, ·〉. Always, V ∗ is a Banach space if endowed by the norm (1.7), denotedoften briefly ‖ · ‖∗ instead of ‖ · ‖V ∗ , i.e. ‖f‖∗ = sup‖v‖≤1〈f, v〉. Obviously,

〈f, u〉 = ‖u‖⟨f,

u

‖u‖⟩≤ ‖u‖ sup

‖v‖≤1

〈f, v〉 = ‖f‖∗ ‖u‖. (1.8)

6A “sphere” {v∈V ; ‖v‖ = �} (of the radius � > 0) is a surface of a “ball” {v∈V ; ‖v‖ ≤ �}.7The property (1.6) is often called sequential lower semicontinuity.8The conventional norm on R is the absolute value | · |.


If V is a Hilbert space, then (u �→ (f, u)) ∈ V ∗ for any f ∈ V , and the mappingf �→ (u �→ (f, u)) identifies V with V ∗. Then (1.8) turns into a so-called Cauchy-Bunyakovskiı inequality (f, u) ≤ ‖f‖ ‖u‖.

Having two Banach spaces V1, V2, we have(V1 ∩ V2

)∗ ∼= V ∗1 + V ∗2 :={f = f1+f2; f1∈V ∗1 , f2∈V ∗2

}(1.9)

if the duality pairing is defined by 〈f, v〉 = 〈f1, v〉V ∗1 ×V1 + 〈f2, v〉V ∗

2 ×V2 with f =f1 + f2, and if the norm on V1 ∩ V2 is taken as ‖v‖V1∩V2 := max(‖v‖V1 , ‖v‖V2)while ‖f‖V ∗

1 +V ∗2:= inff=f1+f2(‖f1‖V ∗

1+ ‖f2‖V ∗

2).

Theorem 1.1 (Banach-Steinhaus principle [36]). Let {Aα}α∈S be a family inL (V1, V2), V1 a Banach space, V2 a normed linear space. Then boundedness of{Aα(v)}α∈S ⊂ V2 for any v∈V1 implies boundedness of {Aα}α∈S ⊂ L (V1, V2).

One can consider a normed linear space V equipped with the collection ofseminorms {v �→ |〈f, v〉|}f∈V ∗ , which makes it a locally convex space. Sequencesconverging in this locally convex space will be called weakly convergent. Any se-quence {uk}k∈N converging in the original norm, i.e. limk→∞ ‖uk − u‖V = 0, willbe called norm-convergent or also strongly convergent and is automatically weaklyconvergent.9

If the convergence in (1.6) refers to the weak one, the function f : V → Rwill be called weakly lower (resp. upper) semicontinuous.

Theorem 1.2. 10 If V is uniformly convex, uk ⇀ u, and ‖uk‖ → ‖u‖, then uk → u.

Likewise, also the Banach space V ∗ can be endowed by the collection ofseminorms {f �→ |〈f, v〉|}v∈V , which makes it a locally convex space. Sequencesconverging in this locally convex space will be called weakly* convergent. Anysequence {fk}k∈N converging in the original norm, i.e. limk→∞ ‖fk − f‖∗ =0, is automatically weakly* convergent. The duality pairing is continuous ifV ∗ × V is equipped with weak*×norm or norm×weak topology11 and separately(weak*,weak)-continuous.12

Proposition 1.3. If V ∗ is separable, then so is also the space V .

The Banach-Steinhaus Theorem 1.1 has immediately the following

Corollary 1.4. Every weakly* convergent sequence in V ∗ must be bounded if Vis a Banach space. In particular, every weakly convergent sequence in a reflexiveBanach V must be bounded.

9The converse implication holds only if the normed linear space V is finite-dimensional, i.e. iso-metrically isomorphic to an Euclidean space.

10See Fan and Glicksberg [140] for thorough investigation and various modifications.11This means limk→∞〈fk, uk〉 = 〈f, u〉 if either fk → f weakly* and uk → u strongly or

fk → f strongly and uk → u weakly.12This means both limk→∞ liml→∞〈fk , ul〉 = 〈f, u〉 and liml→∞ limk→∞〈fk, ul〉 = 〈f, u〉 if

fk → f weakly* and uk → u weakly.


Having two locally convex spaces V1 and V2 and an operator A ∈ L (V1, V2),we define the so-called adjoint operator A∗ ∈ L (V ∗2 , V

∗1 ) by the identity 〈A∗f, v〉 =

〈f,Av〉 to be valid for any v ∈ V1 and f ∈ V ∗2 . If V1 and V2 are normed linearspaces, then A �→ A∗ realizes an isometrical (i.e. norm-preserving) isomorphismbetween L (V1, V2) and L (V ∗2 , V

∗1 ).

Besides, forgetting that V ∗ has been created by its predual V , one can thinkabout the weak convergence on V ∗, induced by the collection of seminorms {f �→|〈φ, f〉|}φ∈(V ∗)∗ . The space V ∗∗ := (V ∗)∗ is called a bidual to V and the spaceV itself is embedded into it by the canonical embedding i : V → V ∗∗ definedby 〈i(v), f〉 = 〈f, v〉. We will often identify13 V with its image i(V ) in V ∗∗ sothat always V ⊂ V ∗∗, and therefore any weakly convergent sequence in V ∗ is alsoweakly* convergent (to the same limit). The converse implication holds if andonly if V ≡ i(V ) = V ∗∗, at which case the Banach space V is called reflexive. TheMilman-Pettis theorem [285, 335] says that every uniformly convex Banach spacemust be reflexive. The Asplund theorem [23] says that any reflexive Banach spacecan equivalently be renormed14 so that it is, together with its dual, strictly convex.By the Clarkson theorem [97], also separable Banach spaces can be renormed tobe strictly convex.15

A mapping A : V1 → V2 is called bounded if it maps bounded sets in V1

into bounded sets in V2. The normed structure of V1 and V2 allows us to definea mapping A : V1 → V2 to be Lipschitz continuous if, for some ∈ R and allu, v ∈ V , it holds that ‖A(u) − A(v)‖V2 ≤ ‖u − v‖V1 ; in this case, is called aLipschitz constant. Obviously, any Lipschitz continuous mapping is bounded anduniformly continuous in the sense that ‖A(u) − A(v)‖V2 ≤ ζ(‖u − v‖V1) with ζincreasing, ζ(0) = 0. If ≤ 1 (resp. < 1), A is called non-expansive (resp. acontraction).

The linear structure of V1 and V2 allows us to investigate smoothness of A.We say that A : V1 → V2 has the directional derivative at u ∈ V in the directionh ∈ V , denoted by DA(u, h), if there is the limit

limε↘0

A(u+ εh)−A(u)

ε=: DA(u, h). (1.10)

If the mapping h �→DA(u, h) is linear and continuous, then we say that A has aGateaux [173] differential at u ∈ V , denoted by A′(u) ∈ L (V1, V2). If the Gateauxdifferential exists in any point, A is called Gateaux differentiable and A′ : V �→L (V1, V2). In the special case V2 = R, a Gateaux-differentiable functional Φ :V1 → R has the differential Φ′ : V1 → V ∗1 .

13This identification is indeed very natural because the mapping i : V → V ∗∗ realizes a(weak,weak*)- as well as (norm,norm)-homeomorphical embedding.

14Two norms ‖ · ‖1 and ‖ · ‖2 on V are called equivalent to each other if ∃ε > 0 ∀v ∈ V :ε‖v‖1 ≤ ‖v‖2 ≤ ε−1‖v‖1.

15These renormalization results can be still improved for so-called locally uniformly convexspaces, i.e. δ in (1.5) depends on u; cf. Troyanski [412].


1.1.3 Convex sets

A set A in a linear space is called convex if λu + (1−λ)v ∈ A whenever u, v ∈A and λ ∈ [0, 1], and it is called a cone (with the vertex at the origin 0) ifλv ∈ A whenever v ∈ A and λ ≥ 0. A function f : V → R is called convex iff(λu + (1−λ)v) ≤ λf(u) + (1−λ)f(v) for any u, v ∈ V , λ ∈ [0, 1]. If u �= v andλ ∈ (0, 1) imply a strict inequality, then f is called strictly convex. A functionf : V → R is convex (resp. lower semicontinuous) if and only if its epigraphepi(f) := {(x, a) ∈ V × R; a ≥ f(x)} is a convex (resp. closed) subset of V × R.

Theorem 1.5 (Hahn [196] and Banach [34]). Let K be an open convex nonemptysubset of a locally convex space V and L be a linear manifold that does not intersectK. Then there is a closed hyperplane L such that L ⊂ L and K ∩ L = ∅. In otherwords, there is f ∈ V ∗ such that 〈f, u〉 > 〈f, v〉 whenever u ∈ K and v ∈ L.

Proposition 1.6. A closed set A is convex if and only if 12u + 1

2v ∈ A wheneveru, v ∈ A. Closed convex sets are weakly closed.

This is related to the fact that a lower semicontinuous functional f is convexif and only if

1

2f(u1) +

1

2f(u2) ≥ f

(u1 + u2

2

), (1.11)

cf. Exercise 4.18 below.Having a convex subset K of a locally convex space V and u ∈ K, we define

the tangent cone TK(u) ⊂ V by

TK(u) := cl( ⋃

a>0

a(K−u)). (1.12)

Obviously, TK(u) is a closed convex cone and v ∈ TK(u) means precisely thatu + akvk ∈ K for suitable sequences {ak}k∈N ⊂ R and {vk}k∈N ⊂ V such thatlimk→∞ vk = v. Besides, we define the normal cone NK(v) as

NK(u) :={f ∈V ∗; ∀v∈TK(u) : 〈f, v〉 ≤ 0

}. (1.13)

Again, the normal cone is always a closed convex cone in V ∗.

TK(u) TK(u)

NK(u)

NK(u)

K

Figure 1. Two examples of the tangent and the normal cones to a convex set K⊂R2

≡ (R2)∗; for illustration, the cones are shifted to the respective points u’s.


1.1.4 Compactness

An important notion on which a lot of powerful tools are based is compactness. Letus agree, for a certain simplicity16, to say that a set V in a locally convex spaceV is compact if every sequence in A contains a convergent subsequence whoselimit belongs to A. Having in mind a Banach space V (possibly having a predual)with a structure of the norm (resp. weak or possibly weak*) locally convex space,we will specify compactness by the adjective “norm” (resp. “weak” or possibly“weak*”); if no adjective is mentioned, then the “norm” one will implicitly beunderstood. A weakening of the notion of compactness is precompactness: we saythat a set A is precompact in the sense that every sequence in such a set containsa Cauchy subsequence. Another modification is the following: we say that A isrelatively compact if its closure is compact. The reader can easily guess whate.g. “relatively weakly* compact” or “relatively norm compact” mean. If everyCauchy sequence converges (in particular in a Banach space), the prefix/adjective“pre-” and “relatively” coincide with each other. Thus, in a Banach space, relativenorm compactness and norm precompactness are the same.

A mapping A : V1 → V2, V1, V2 Banach spaces, is called totally continu-ous if it is (weak,norm)-continuous, i.e. it maps weakly convergent sequences tostrongly convergent ones. It is called compact if it maps bounded sets in V1 intoprecompact17 sets in V2. If V1 is reflexive, then any totally continuous mapping iscompact18 but not conversely19.

Theorem 1.7 (Selection principle, Banach [35, Chap.VIII, Thm.3]20). In aBanach space with a separable predual, any bounded sequence contains a weakly*convergent subsequence.

Often, Theorem 1.7 is also called, not completely exactly however, Alaoglu-Bourbaki’s theorem21.

16In fact, our “compactness” is in general topology called “sequential compactness” whilecompactness means that every net (=a generalized sequence) contains a convergent finer net(=a suitable generalization of the notion of “subsequence”), or, equivalently, that every coveringof A by open sets contains a finite sub-covering. These two concepts coincide with each otherin a lot of important cases, primarily in normed spaces (considering norm compactness). Moregenerally, it happens if the structure of a locally convex spaces can be equally induced by acountable collection of seminorms {| · |ξ}ξ∈N, e.g. it concerns weak (resp. weak*) compactness

if V has a separable dual (resp. predual). A far less trivial fact, known as the Eberlein-Smuljantheorem, is that in a Banach space the relatively weakly compact (in the general-topology sense)sets coincide with the relatively weakly sequentially compact ones.

17As V2 is assumed a Banach space, precompact sets are just those which are relatively com-pact.

18For B ⊂ V1 bounded, any sequence in A(B), say {A(vk)}k∈N with vk ∈ B contains asubsequence convergent in V2, e.g. {A(vkl

)}l∈N with {vkl}l∈N weakly convergent in V1; here

both reflexivity of V1 guaranteeing existence of {vkl}l∈N and compactness of A guaranteeing

convergence of {A(vkl)}l∈N has been used.

19See Remark 2.39 and Exercise 2.64 below.20For the proof cf. Exercise 2.51.21In fact, Alaoglu-Bourbaki’s theorem says that the polar set to a neighbourhood of the origin

in a locally convex space is weakly* compact, see Alaoglu [13] and Bourbaki [63].


Theorem 1.8 (Bolzano and Weierstrass22). Every lower (resp. upper) semi-continuous function X → R on a compact set attains its minimum (resp. maxi-mum) on this set.

1.1.5 Fixed-point theorems

A point u ∈ V is called a fixed point of a mapping M : V → V if M(u) = u.

Theorem 1.9 (Schauder fixed-point theorem [379]). A continuous compactmapping on a closed, bounded, convex set in a Banach space has a fixed point.Alternatively, a continuous mapping on a compact, convex, set in a Banach spacehas a fixed point.

As a special case for V = Rn, one gets a historically older achievement whichhas been tremendously generalized by the previous Theorem 1.9:

Theorem 1.10 (Brouwer fixed-point theorem [72]). A continuous mappingon a compact convex set in Rn has a fixed point.

Further generalization of Theorem 1.9 is very useful in applications. A usefulconcept of a set-valued mapping M : V ⇒ V just means that M : V → 2V , theset of all subsets of V . Such M is called upper semicontinuous if it has a closedgraph, i.e. uk → u, fk → f , and fk ∈ M(uk) implies f ∈ M(u).

Theorem 1.11 (Kakutani fixed-point theorem [224]23). An upper semicon-tinuous set-valued mapping M : V ⇒ V with nonempty closed convex values whichmaps a nonempty compact convex set K in a locally convex space into itself has afixed point, i.e. ∃u ∈ K: u ∈ M(u).

A completely different principle is based on metric properties and exploitscompleteness instead of compactness:

Theorem 1.12 (Banach fixed-point theorem [33]24). A contractive mappingon a Banach space has a fixed point which is even unique.

1.2 Function spaces

We consider the Euclidean space Rn, n ≥ 1, endowed with standard Euclideantopology and for Ω a subset of Rn we will define various spaces of functions Ω →

22In fact, this is rather a tremendous generalization of the original assertion by Bolzano [60]who showed, rather intuitively (because completion of irrational numbers by transcendental oneswhich locally compactifies R has been discovered only much later) that any real continuousfunction of a bounded closed interval is bounded. An essence, called the Bolzano-Weierstrassprinciple, is that every sequence in a compact set has a cluster point, i.e. a point whose eachneighbourhood contains infinitely many members of this sequence.

23The original version [224] formulated in Rn has been generalized for locally convex spacesby Fan [139] and Glicksberg [181].

24In fact, this theorem works (and was formulated in [33]) in a complete metric space, too.

1.2. Function spaces 9

Rm. If endowed by a pointwise addition and multiplication25 the linear-space struc-ture of Rm is inherited by these spaces. Besides, we will endow them by norms,which makes them normed linear (or, mostly even Banach) spaces. Having twosuch spaces U ⊂ V , we say that the mapping I : U → V : u �→ u is a continuousembedding (or, that U is embedded continuously to V ) if the linear operator I iscontinuous (hence bounded). This means that ‖u‖V ≤ N‖u‖U ; for N one can takethe norm ‖I‖L (U,V ). If I is compact, we speak about a compact embedding and usethe notation U � V . If U is a dense subset in V , we will speak about a dense em-bedding; this property obviously depends on the norm of V but not of U . It followsby a general functional-analysis argument that the adjoint mapping I∗ : V ∗ → U∗

is continuous and injective provided U ⊂ V continuously and densely26; then wecan identify V ∗ as a subset of U∗.

1.2.1 Continuous and smooth functions

The notation C(·), C0(·), and C0,1(·) will indicate sets of continuous, boundedcontinuous, and Lipschitz continuous functions, respectively.27 E.g. C0(Ω;Rm)denote the set of bounded continuous functions Ω → Rm. We denote by Ω theclosure of Ω in the Euclidean space Rn. By the Bolzano-Weierstrass Theorem 1.8,C0(Ω;Rm) = C(Ω;Rm) if Ω is bounded. Equally, we can understand C(Ω;Rm) ascomposed from functions Ω → Rm having a continuous extension on the closure Ω.For m = 1, we will write briefly C0(Ω) (resp. C0,1(Ω)). If equipped by a pointwiseaddition and multiplication and by the norm

‖u‖C0(Ω;Rm) := maxx∈Ω

|u(x)|, ‖u‖C0,1(Ω;Rm) := supx,ξ∈Ωx �=ξ

(|u(x)|+ |u(x)− u(ξ)|

|x− ξ|)

with |·| denoting the Euclidean norm in the corresponding spaces, both C0(Ω;Rm)and C0,1(Ω;Rm) become Banach spaces.

Furthermore, for k ≥ 1, we define spaces of smooth functions, having deriva-tives up to k-th order continuous up to the boundary, i.e.

Ck(Ω;Rm) :={u ∈ C0(Ω;Rm); ∀(i1, . . . , in) ∈ (N ∪ {0})n,n∑

α=1

iα ≤ k :∂i1+···+inu

∂i1x1 · · · ∂inxn∈ C0(Ω;Rm)

}. (1.14)

If endowed by the linear structure of C0(Ω;Rm) and by the norm ‖u‖Ck(Ω;Rm) :=∥∥u∥∥C0(Ω;Rm)

+∑

i1+···+in≤k

∥∥ ∂i1+···+in

∂i1x1···∂inxnu∥∥C0(Ω;Rm)

, they become Banach spaces.

25This means, for u, v : Ω → Rm and λ ∈ R, we define [u + v](x) := u(x) + v(x) and[λu](x) := λu(x) for all x∈ Ω.

26Indeed, I∗ is injective (because two different linear continuous functionals on V must havealso different traces on any dense subset, in particular on U).

27For Ω = Rn, the adjectives “continuous” and “Lipschitz continuous” can be understood asin Section 1.1 because both Rn and Rm are normed spaces, while for a general Ω it is just arestriction of these definitions on Ω.


The continuous embedding Ck(Ω;Rm) ⊂ Cl(Ω;Rm) holds for any k ≥ l ≥ 0,i.e. ‖u‖Cl(Ω;Rm) ≤ N‖u‖Ck(Ω;Rm). In this particular case,N = 1. This embedding iseven dense and, if k > l ≥ 0, also compact. As functions with bounded derivativesare Lipschitz continuous, we have also C1(Ω;Rm) ⊂ C0,1(Ω;Rm).

By Riesz’s theorem, the dual space C0(Ω)∗ is isometrically isomorphic withthe space M (Ω) of regular Borel measures on Ω; a measure is a σ-additive setfunction28 and a regular Borel measure μ ∈ M (Ω) is a measure on the so-called Borel σ-algebra29 which is regular30 and has a finite variation31 |μ| overΩ, i.e. |μ|(Ω) < +∞. The mentioned isomorphism f �→ μ : C0(Ω)∗ → M (Ω) is by〈f, v〉 = ∫

Ωv μ(dx).32 Then ‖f‖C0(Ω)∗ = |μ|(Ω). For x ∈ Ω, a measure δx ∈ M (Ω)

defined by 〈δx, v〉 = v(x) is called Dirac’s measure supported at x.Considering an increasing sequence of compact subsets Ki ⊂ Ω such that

Ω =⋃

i∈N Ki, we put

D(Ω):=⋃i∈N

⋂k∈N

CkKi

(Ω), (1.15)

where CkKi

(Ω) denotes the space of all functions Ω → R which are continuoustogether with all their derivatives up to the order k and which have the supportcontained in Ki. Each C∞Ki

:=⋂

k∈N CkKi

(Ω) is endowed by the collection of semi-

norms{| · |k,Ki

}k∈N with |u|k,Ki :=

∥∥u|Ki

∥∥Ck(Ki)

, which makes it a locally convex

space. Then D(Ω) =⋃

i∈N C∞Kiis equipped with the finest topology that makes

all the embeddings C∞Ki→ D(Ω) continuous,33 which makes it a locally convex

space. The elements of the dual space D(Ω)∗ are called distributions on Ω.

1.2.2 Lebesgue integrable functions

The n-dimensional outer Lebesgue measure measn(·) on the Euclidean space Rn,n ≥ 1, is defined as

measn(A) := inf{ ∞∑

k=1

n∏i=1

bki−aki : A ⊂∞⋃k=1

[ak1 , bk1 ]× · · · ×[akn, b

kn], aki≤bki

}(1.16)

and then we call a set A ⊂ Rn Lebesgue measurable if measn(A) = measn(A∩S)+measn(A \ S) for any subset S ⊂ Rn.34 The collection Σ of Lebesgue measurable

28This means μ(⋃

i∈NAi) =

∑i∈N

μ(Ai) for any mutually disjoint Ai from an algebra inquestion.

29A collection Σ of subsets of Ω is called a σ-algebra if Ai∈Σ ⇒ ⋃i∈N

Ai∈Σ, ∅∈Σ, and

A∈Σ⇒ Ω\A∈Σ. Borel’s σ-algebra is the smallest σ-algebra containing all open subsets of Ω.30A set function μ is regular if ∀A∈Σ ∀ε>0 ∃A1, A2∈Σ: cl(A1)⊂A⊂int(A2) and |μ|(A2\A1)≤ε.31For μ additive, we define the variation |μ| of μ by |μ|(A) = sup(A1,...,AI)∈M(A)

∑Ii=1 |μ(Ai)|,

where M(A) denotes the set of all finite collections (A1, . . . , AI) of mutually disjoint Ai ∈ Σsuch that Ai ⊂ A for any i = 1, . . . , I.

32The integral via μ is defined by limit of simple functions similarly as in Section 1.2.2.33The convergence fk → f in D(Ω) means that ∃K ⊂ Ω compact ∃k0∈N ∀k ≥ k0: supp(fk) ⊂

K and fk → f in ClK(Ω) for any l∈N; cf. e.g. [425, Sect.I.1].

34For example, all closed sets are Lebesgue measurable, hence every open set too, as well astheir countable union or intersection, etc.


subsets of Ω forms a so-called σ-algebra35 which, together with the functionmeasn : Σ → R ∪ {+∞}, have (and are characterized by) the following prop-erties:1. A open implies A ∈ Σ,

2. A = [a1, b1]× · · · × [an, bn] with ai ≤ bi implies measn(A) =∏n

i=1(bi − ai),

3. measn is countably additive, i.e. measn(⋃

k∈N Ak

)=∑

k∈Nmeasn(Ak) for anycountable collection {Ak}k∈N of pairwise disjoint sets Ai ∈ Σ,

4. A ⊂ B ∈ Σ and measn(B) = 0 implies A ∈ Σ and measn(A) = 0.

The function measn : Σ → R ∪ {+∞} is called the Lebesgue measure. Having aset Ω ∈ Σ, we say that a property holds almost everywhere on Ω (in abbreviationa.e. on Ω) if this property holds everywhere on Ω with the possible exception of aset of Lebesgue-measure zero; referring to those x where this property holds, wewill also say that it holds at almost all x ∈ Ω (in abbreviation a.a. x ∈ Ω).

A function u : Rn → Rm is called (Lebesgue) measurable if u−1(A) := {x ∈Rn; u(x) ∈ A} is Lebesgue measurable for any A ∈ Rm open.

We call u : Rn → Rm simple if it takes only a finite number of valuesvi ∈ Rm and u−1(vi) = {x; u(x) = vi} ∈ Σ; then we define the integral

∫Rn u(x) dx

naturally as∑

finitemeasn(Ai)vi. Furthermore, a measurable u is called integrableif there is a sequence of simple functions {uk}k∈N such that limk→∞ uk(x) = u(x)for a.a. x ∈ Ω and limk→∞

∫Rn uk(x) dx does exist in R. Then, this limit will

be denoted by∫Rn u(x) dx and we call it the (Lebesgue) integral of u. It is then

independent of the particular choice of the sequence {uk}k∈N.We will now consider Ω ⊂ Rn measurable with measn(Ω) < +∞. The notions

of measurability and the integral of functions Ω → Rm can be understood as beforeprovided all these functions are extended on Rn \ Ω by 0. By Lp(Ω;Rm) we willdenote the set of all measurable functions36 u : Ω → Rm such that ‖u‖Lp(Ω;Rm) <+∞, where37

‖u‖Lp(Ω;Rm) :=

⎧⎨⎩p

√∫Ω |u(x)|pdx for 1 ≤ p < +∞ ,

ess supx∈Ω

|u(x)| for p = +∞ (1.17)

and | · | is the Euclidean norm on Rm. The set Lp(Ω;Rm), endowed by a pointwiseaddition and scalar multiplication, is a linear space. Besides, ‖·‖Lp(Ω;Rm) is a norm

35We call Σ an algebra if ∅ ∈ Σ, A ∈ Σ ⇒ Ω \ A ∈ Σ, and A1, A2 ∈ Σ ⇒ A1 ∪ A2 ∈ Σ.If also Ai ∈ Σ ⇒ ⋃

i∈NAi ∈ Σ, then Σ will be called a σ-algebra. An example is the so-called

Borel σ-algebra: the smallest σ-algebra containing all open subsets of Ω. In fact, Σ is the so-called Lebesgue extension of the Borel σ-algebra, created by adding all subsets of sets havingthe measure zero.

36As usual, we will not distinguish between functions that are equal to each other a.e., so that,strictly speaking, Lp(Ω;Rm) contains classes of equivalence of such functions.

37The “essential supremum” is defined as

ess supx∈Ω

f(x) = infN⊂Ω

measn(N)=0

supx∈Ω\N

f(x).


on Lp(Ω;Rm) which makes it a Banach space, called a Lebesgue space.If a measure μ ∈ M (Ω) possesses a density dμ ∈ L1(Ω), which means μ(A) =∫

A dμ(x) dx for any measurable A ⊂ Ω, then μ has a certain special property,namely it is absolutely continuous with respect to the Lebesgue measure38 andalso the converse assertion is true: every absolutely continuous measure possessesa density belonging to L1(Ω). This fact is known as the Radon-Nikodym theorem.

An important question is how to characterize concretely the dual spaces. The“natural” duality pairing comes from the inner product in L2-spaces, which means〈u, v〉 := ∫

Ω u(x)·v(x) dx, where u ·v :=∑m

i=1 uivi is the inner product in Rm. If1 < p < +∞, then Lp(Ω;Rm) is reflexive. From the algebraic Young inequality

ab ≤ 1

pap +

1

p′bp

′(1.18)

one gets39 the Holder [208] (or, for p=2, the Holder-Bunyakovskiı [85]) inequality:40∫Ω

∣∣u(x)·v(x)∣∣ dx ≤ p

√∫Ω|u(x)|pdx p′

√∫Ω|v(x)|p′

dx, (1.19)

where p′ is the so-called conjugate exponent defined as

p′ :=

⎧⎨⎩p/(p−1) for 1 < p < +∞,

1 for p = +∞,

+∞ for p = 1.

(1.20)

In fact, the modification of (1.19) for p = 1 or p = +∞ looks trivially as∫Ω |u ·

v|dx ≤ ‖u‖L∞(Ω;Rn)‖u‖L1(Ω;Rn). From (1.19), it can be shown that the dual space

is isometrically isomorphic with Lp′(Ω;Rm) if 1 ≤ p < +∞. On the other hand,

the dual space to L∞(Ω;Rm) is substantially larger than L1(Ω;Rm).41 Applyingthe algebraic Young inequality (1.18) to |u(x)·v(x)| ≤ |u(x)| |v(x)| and integratingit over Ω, we obtain another important inequality, the integral Young inequality∫

Ω

∣∣u(x) · v(x)∣∣ dx ≤ 1

p

∫Ω

|u(x)|p dx+1

p′

∫Ω

|v(x)|p′dx. (1.21)

Instead of (1.18), we will often apply the modified Young inequality

∀ε > 0 : ab ≤ εap + Cεbp′, where Cε :=

p−1√ε p′√p

p− 1. (1.22)

Moreover, for 1 < p < +∞, the space Lp(Ω;Rm) is uniformly convex.42

38This means that ∀ε > 0 ∃δ > 0 ∀A ⊂ Ω measurable: measn(A) ≤ δ =⇒ |μ(A)| ≤ ε.39Hint: ‖u‖−1

Lp(Ω)‖v‖−1

Lp′(Ω)

∫Ω |u · v|dx ≤ 1

p‖u‖−p

Lp(Ω)

∫Ω |u|pdx+ 1

p′ ‖v‖−p′

Lp′(Ω)

∫Ω |u|p

′dx = 1.

40Originally, Holder [208] states this in a less symmetrical form for sums in place of integrals.41The elements of L∞(Ω;Rm)∗ are indeed very abstract objects and can be identified with

finitely-additive measures vanishing on zero-measure sets, see Yosida and Hewitt [426].42This result is due to Clarkson [97], see also e.g. Adams [3, Corollary 2.29] or Kufner at

al. [245, Remark 2.17.8].


If measn(Ω) < +∞ and 1 ≤ q ≤ p ≤ +∞, the embeddings C0(Ω;Rm) ⊂Lp(Ω;Rm) ⊂ Lq(Ω;Rm) are continuous43. Moreover, for p < +∞ these embed-dings are dense and then, as C0(Ω;Rm) is separable, Lp(Ω;Rm) is separable, too.On the other hand, L∞(Ω) is not separable.44

Holder’s inequality also allows for an interpolation between Lp1(Ω) andLp2(Ω): for p1, p2, p ∈ [1,+∞], λ ∈ [0, 1], it holds that45

1

p=

λ

p1+

1−λ

p2⇒ ‖v‖Lp(Ω) ≤ ‖v‖λLp1(Ω)‖v‖1−λ

Lp2(Ω). (1.23)

Moreover, if p is as in (1.23), p1 ≤ p2, and similarly q−1 = λq−11 + (1−λ)q−1

2 ,q1 ≤ q2, and A is a bounded linear operator Lp1(Ω) → Lq1(Ω) whose restrictionon Lp2(Ω) belongs to L (Lp2(Ω), Lq2(Ω)), then∥∥A∥∥

L (Lp(Ω),Lq(Ω))≤ C

∥∥A∥∥λL (Lp1(Ω),Lq1 (Ω))

∥∥A∥∥1−λ

L (Lp2(Ω),Lq2 (Ω))(1.24)

for some constant C = C(p1, p2, q1, q2, λ).We say that a sequence uk : Ω → Rm converges in measure to u if

∀ε > 0 : limk→∞

measn({x∈Ω; |uk(x)−u(x)| ≥ ε}) = 0. (1.25)

Naturally, the convergence a.e. means that uk(x) → u(x) for a.a. x∈Ω. Let us em-phasize that convergence in measure does not imply convergence a.e.46. Anyhow:

Proposition 1.13 (Various modes of convergences).

(i) Any sequence converging a.e. converges also in measure.

(ii) Any sequence converging in measure admits a subsequence converging a.e.

(iii) Any sequence converging in L1(Ω) converges in measure.

Theorem 1.14 (Lebesgue [254]). Let {uk}k∈N ⊂ L1(Ω) be a sequence converginga.e. to some u and |uk(x)| ≤ v(x) for some v ∈ L1(Ω). Then u lives in L1(Ω) andlimk→∞

∫Auk(x) dx =

∫Au(x) dx for any A ⊂ Ω measurable.

Theorem 1.15 (Fatou [143]). Let {uk}k∈N ⊂ L1(Ω) be a sequence of non-negative functions47 such that lim infk→∞

∫Ω uk(x) dx < +∞. Then the function

x �→ lim infk→∞ uk(x) is integrable and

lim infk→∞

∫Ω

uk(x) dx ≥∫Ω

(lim infk→∞

uk(x))dx. (1.26)

43Cf. Exercise 2.68 below.44To see it, consider Ω = (0, 1) and a collection {χ(0,a)}a∈(0,1) of characteristic functions of an

interval (0, a), i.e. χ(0,a)(x) = 1 for x ∈ (0, a) and χ(0,a)(x) = 0 for x ∈ [a, 1). This collection isan uncountable subset of the unit sphere in L∞(0, 1) and ‖χ(0,a)−χ(0,b)‖L∞(0,1) = 1 for a �= b,hence L∞(0, 1) cannot be separable.

45Cf. Exercise 2.59 below. See e.g. [268, p.26].46An example for a sequence converging in the measure on [0, 1] to 0 but not a.e. is u(l,m)

a characteristic function of the interval of the type [(m − 1)2−l ,m2−l] for 1 ≤ m ≤ 2l, ar-ranged lexicographically as (l, m) = (1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), . . .. Selection ofa subsequence converging a.e. to 0 can be done, e.g., by taking m = 1 only.

47Obviously, existence of a common integrable minorant can weaken this assumption.


In particular, Theorem 1.14 says that the set {uk; k ∈ N} is relatively weaklycompact48 in L1(Ω). This property is related to both equi-absolute continuity anduniform integrability:

Theorem 1.16 (Dunford and Pettis [128]). Let M ⊂ L1(Ω;Rm) be bounded.Then the following statements are equivalent to each other:(i) M is relatively weakly compact in L1(Ω;Rm),

(ii) the set M is uniformly integrable, which means:

∀ε > 0 ∃K ∈ R+ : supu∈M

∫{x∈Ω;|u(x)|≥K}

|u(x)| dx ≤ ε , (1.27)

(iii) the set M is equi-absolutely-continuous, which means:

∀ε > 0 ∃δ > 0 : supu∈M

sup|A|≤δ

∫A

|u(x)| dx ≤ ε . (1.28)

Useful generalizations of the Lebesgue dominated-convergence Theorem 1.14and the Fatou Theorem 1.15 are:

Theorem 1.17 (Vitali [421]49). Let {uk}k∈N ⊂ L1(Ω) be a sequence converginga.e. to some u. Then u ∈ Lp(Ω) and uk → u in Lp(Ω) if and only if {|uk|p}k∈N isuniformly integrable.

Theorem 1.18 (Fatou, generalized50). The conclusion of Theorem 1.15 holdsif uk ≥ 0 is replaced by uk ≥ vk with {vk}k∈N being uniformly integrable.

Theorem 1.19 (Fubini [160]). Considering two Lebesgue measurable sets Ω1 ⊂Rn1 and Ω2 ⊂ Rn2 , the following identity holds provided g ∈ L1(Ω1×Ω2) (inparticular, each of the following double-integrals does exist and is finite):∫Ω1×Ω2

g(x1, x2) d(x1, x2) =

∫Ω1

(∫Ω2

g(x1, x2) dx2

)dx1 =

∫Ω2

(∫Ω1

g(x1, x2) dx1

)dx2.

1.2.3 Sobolev spaces

Modern theory of differential equations is based on spaces of functions whosederivatives exist in a generalized sense and enjoy a suitable integrability. Thosespaces, developed since the works by Beppo Levi [258], Leray [256], Sobolev [389],

48Note that the linear hull of all characteristic functions χA with A ⊂ Ω measurable is densein L∞(Ω) ∼= L1(Ω)∗, so that the sequence {uk}, being bounded in L1(Ω), converges weakly inL1(Ω) and, as such, it is relatively sequentially weakly compact, hence by the Eberlein-Smuljantheorem relatively weakly compact, too.

49More precisely, in [421], the integration of summable series is investigated rather than meresequences.

50See Ash [21, Thm.7.5.2], or also Klei and Miyara [233] or Saadoune and Valadier [377].


and Tonelli [406], have thus an absolute importance in this context and theirtheory is presently very broad. We present here only a rather minimal extent; formore detailed exposition we refer to Adams [3], Adams, Fournier [4], Giusti [180],Kufner, Fucık, John [245], Maz’ya [277], or Ziemer [430].

Having a function u ∈ Lp(Ω), we define its distributional derivative∂ku/∂xk1

1 · · · ∂xknn with k1+ · · ·+kn = k and ki ≥ 0 for any i = 1, . . . , n as a

distribution such that

∀g∈D(Ω) :⟨ ∂ku

∂xk11 · · · ∂xkn

n

, g⟩= (−1)k

∫Ω

u∂kg

∂xk11 · · · ∂xkn

n

dx. (1.29)

The n-tuple of the first-order distributional derivatives(

∂∂x1

u, . . . , ∂∂xn

u)is de-

noted by ∇u and called a gradient of u. We will now consider Ω ⊂ Rn open withmeasn(Ω) < +∞; such a set will be called a domain. For p < +∞, we define aSobolev space [390]

W 1,p(Ω):={u∈Lp(Ω); ∇u∈Lp(Ω;Rn)

}, equipped with the norm (1.30a)

‖u‖W 1,p(Ω) :=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩p

√‖u‖pLp(Ω) + ‖∇u‖pLp(Ω;Rn)

= p

√∫Ω

∣∣u(x)∣∣p + ∣∣∇u(x)∣∣pdx if p < +∞,

max(‖u‖L∞(Ω), ‖∇u‖L∞(Ω;Rn)

)= ess supx∈Ωmax

(|u(x)|, |∇u(x)|) if p = +∞.

(1.30b)

As, by Rademacher’s theorem, Lipschitz functions are a.e. differentiable, it holdsthat W 1,∞(Ω) = C0,1(Ω).

Analogously, for k > 1 integer, we define

W k,p(Ω) :={u∈Lp(Ω;Rm); ∇ku∈Lp(Ω;Rnk

)}, (1.31)

where ∇ku denotes the set of all k-th order partial derivatives of u understoodin the distributional sense. The standard norm on W k,p(Ω) is ‖u‖Wk,p(Ω) =(‖u‖pLp(Ω) + ‖∇ku‖p

Lp(Ω;Rnk )

)1/p, which makes it a Banach space. Likewise for

Lebesgue spaces, for 1 ≤ p < +∞ the Sobolev spaces W k,p(Ω;Rm) are separa-ble and, if 1 < p < +∞, they are uniformly convex,51 hence by Milman-Pettis’theorem also reflexive.

The spaces of Rm-valued functions, W k,p(Ω;Rm), are defined analogously.52

To give a good sense to traces on the boundary Γ := ∂Ω := Ω \ Ω, we mustqualify Ω suitably. We say that Ω is a Lipschitz domain if there is a finite numberof overlapping parts Γi of the boundary of Γ and corresponding coordinate systems(i.e. transformation unitary matrices Ai and open sets Gi ∈ Rn−1) such that each

51See Adams [3, Theorem 3.5].52This means W k,p(Ω;Rm) := {(u1, . . . , um); ui ∈ W k,p(Ω)}.


Γi can be expressed as a graph of a Lipschitz function gi ∈ C0,1(Rn−1) in thesense that

Γi ={Aiξ; ξ∈Rn, (ξ1, . . . , ξn−1)∈Gi, ξn = gi(ξ1, . . . , ξn−1)

}(1.32)

and Ω lies on one side of Γ in the sense that{Aiξ; ξ ∈ Rn, (ξ1, . . . , ξn−1) ∈

Gi, gi(ξ1, . . . , ξn−1)−ε < ξn < gi(ξ1, . . . , ξn−1)} ⊂ Ω and simultaneously{

Aiξ; ξ ∈Rn, (ξ1, . . . , ξn−1)∈Gi, gi(ξ1, . . . , ξn−1) < ξn < gi(ξ1, . . . , ξn−1)+ε} ⊂

Rn \ Ω for some ε> 0. For an example of a Lipschitz domain Ω ⊂ R2, see Figure 2awhere three indicated coordinate systems are sufficient to cover Γ, and Figure 2bwhere this condition fails in a variety of ways.

Ω

ΩΩ

Fig. 2. a) Example of a Lipschitz do-main whose boundary canbe covered by three charts.

b) Examples of a 2D and a 3D domainswhich are not Lipschitz because of 3 and4 spots, respectively (marked by circles).

If the mappings gi belong to Ck,α(Rn−1), we say that the domain Ω is of Ck,α-class. Lipschitz domains are then naturally called of the C0,1-class. We will alwaysassume Ω to be Lipschitz and bounded.

Theorem 1.20 (Sobolev embedding [389]). The continuous embedding

W 1,p(Ω) ⊂ Lp∗(Ω) (1.33)

holds provided the exponent p∗ is defined as

p∗ :=

⎧⎪⎨⎪⎩np

n− pfor p < n ,

an arbitrarily large real for p = n ,+∞ for p > n .

(1.34)

Theorem 1.21 (Rellich, Kondrachov [348, 236]53). The compact embedding

W 1,p(Ω) � Lp∗−ε(Ω) , ε ∈ (0, p∗−1], (1.35)

holds for p∗ from (1.34).

53The pioneering Rellich’s work deals with p = 2 only.


For higher-order Sobolev spaces, one gets, e.g., W 2,p(Ω) ⊂ W 1,np/(n−p)(Ω)by applying Theorem 1.20 on first derivatives, and applying Theorem 1.20 onceagain for np/(n − p) instead of p one comes to W 1,np/(n−p)(Ω) ⊂ Lnp/(n−2p)(Ω)provided 2p < n. It identifies p∗∗ := (p∗)∗ as np/(n−2p) if 2p < n. Proceedingfurther by induction, one comes to:

Corollary 1.22 (Higher-order Sobolev embedding54).

(i) If kp < n, the continuous embedding W k,p(Ω) ⊂ Lnp/(n−kp)(Ω) and the com-pact embedding W k,p(Ω) � Lnp/(n−kp)−ε(Ω) hold for any ε ∈ (0, np

n−kp − 1].

(ii) For kp = n, it holds that W k,p(Ω) � Lq(Ω) for any q < +∞.

(iii) For kp > n, it holds that W k,p(Ω) � C(Ω).

Theorem 1.23 (Trace operator55). There is exactly one linear continuous op-erator T : W 1,p(Ω) → L1(Γ) such that, for any u ∈ C1(Ω), it holds that Tu = u|Γ(=the restriction of u on Γ). Moreover, T remains continuous (resp. is compact)as the mappings

u �→ u|Γ : W 1,p(Ω) → Lp#

(Γ), resp., (1.36a)

u �→ u|Γ : W 1,p(Ω) → Lp#−ε(Γ) , ε ∈ (0, p#−1], (1.36b)

provided the exponent p# is defined as

p# :=

⎧⎪⎨⎪⎩np− p

n− pfor p < n ,

an arbitrarily large real for p = n ,+∞ for p > n .

(1.37)

We will call the operator T from Theorem 1.23 the trace operator, andwrite simply u|Γ instead of Tu even if u ∈ W 1,p(Ω) \ C1(Ω). Then we de-fine W 1,p

0 (Ω) := {v ∈ W 1,p(Ω); v|Γ = 0}. For k > 1, we define similarly

W k,p0 (Ω) := {v ∈ W k,p(Ω); ∇iv ∈ W k−i,p

0 (Ω;Rni

), i = 0, . . . , k − 1}.Another useful result allows for a certain interpolation between the Sobolev

space W k,p(Ω) and the Lebesgue space Lq(Ω):

Theorem 1.24 (Gagliardo-Nirenberg inequality [161, 312, 313]). Let β =β1 + · · ·+ βn, β1, . . . , βn ∈ N ∪ {0}, k ∈ N, r, q, and p satisfy

1

r=

β

n+ λ

(1p− k

n

)+ (1− λ)

1

q,

β

k≤ λ ≤ 1, 0 ≤ β ≤ k − 1, (1.38)

54In fact, the point (iii) can be improved. E.g. for k = 1 and Ω smooth, one has W 1,p(Ω) ⊂C0,α(Ω) with α = 1 − n/p, which is referred to a Morrey inequality. In general, W k,p(Ω) ⊂Ck−[n/p]−1,α(Ω) with α < 1 if n/p ∈ N or α = [n/p] + 1− n/p if n/p �∈ N.

55In fact, u �→ u|Γ : W 1,p(Ω) → W 1−1/p,p(Γ), where the Sobolev-Slobodeckiı spaceW 1−1/p,p(Γ) is defined as in (1.42) but on an (n−1)-dimensional manifold Γ instead ofn-dimensional domain Ω. Then, similarly as in Theorem 1.20, we have the embedding

W 1−1/p,p(Γ) ⊂ Lp# (Γ), resp. W 1−1/p,p(Γ) � Lp#−ε(Γ). For the rather less standard nota-tion of the exponent p#, see e.g. Ciarlet [93, Thm 6.1-7].


then it holds ∥∥∥∥ ∂βv

∂xβ1

1 · · · ∂xβn

1

∥∥∥∥Lr(Ω)

≤ CGN

∥∥v∥∥λWk,p(Ω)

∥∥v∥∥1−λ

Lq(Ω), (1.39)

provided k − β − n/p is not a negative integer (otherwise it holds for λ = |β|/k).For example, if n = 2, then

‖v‖L4(Ω) ≤ CGN

√‖v‖L2(Ω)

(‖v‖L2(Ω) + ‖∇v‖L2(Ω;R2)

). (1.40)

There is another definition of Sobolev spaces, not so explicit as that one weused. Namely, Wk,p(Ω) := cl

(C∞(Ω)

)where “cl” stands for the closure in, say,

Lp(Ω) with respect to the W k,p-norm introduced above; for k = 1 see (1.30b).

Similarly, Wk,p0 (Ω) := cl

(C∞0 (Ω)

)where C∞0 (Ω) stands for the set of infinitely

smooth functions with a compact support in Ω. For the following important as-sertion an important (here permanently accepted) assumption is that Ω has aLipschitz boundary.

Theorem 1.25 (Density of smooth functions). It holds that

W k,p(Ω) = Wk,p(Ω), W k,p0 (Ω) = Wk,p

0 (Ω). (1.41)

A generalization of Sobolev spaces for k ≥ 0 a non-integer is often useful forvarious finer investigations:

W k,p(Ω) :=

{u∈W [k],p(Ω);

∫Ω×Ω

|∇[k]u(x)−∇[k]u(ξ)|p|x− ξ|n+p(k−[k])

dxdξ < +∞}, (1.42)

where [k] denotes the integer part of k. For k non-integer, W k,p(Ω) is called theSobolev-Slobodeckiı space. They are Banach spaces if normed by the norm

‖u‖Wk,p(Ω) :=

(‖u‖p

W [k],p(Ω)+

∫Ω×Ω

|∇[k]u(x)−∇[k]u(ξ)|p|x− ξ|n+p(k−[k])

dxdξ

)1/p

. (1.43)

In fact, Corollary 1.22 holds for k ≥ 0 non-integer, too. A complete interpolationtheory holds for the Hilbertian case p = 2 for general k, k1, and k2 non-negative:

k = λk1 + (1−λ)k2 =⇒ ∥∥u∥∥Wk,p(Ω)

≤ C∥∥u∥∥λ

Wk1,p(Ω)

∥∥u∥∥1−λ

Wk2 ,p(Ω)(1.44)

with some C = C(k1, k2, λ), while for p �= 2, the inequality (1.44) holds providedk is not integer and k1, k2 ∈ N.56 Like in (1.24),∥∥A∥∥

L (Wk,p(Ω),W l,p(Ω))≤ C

∥∥A∥∥λL (Wk1,p(Ω),W l1,p(Ω))

∥∥A∥∥1−λ

L (Wk2,p(Ω),W l1,p(Ω))

(1.45)

56For k ∈ N the Sobolev space W k,p(Ω) is to be replaced by a so-called Besov space.

1.3. Nemytskiı mappings 19

for some constant C = C(k1, k2, l1, l2, λ) provided A ∈ L(W k1,2(Ω),W l1,2(Ω)

) ∩L(W k2,2(Ω),W l2,2(Ω)

), k is as in (1.44), and l−1 = λl−1

1 + (1−λ)l−12 .

The Sobolev-Slobodeckiı spaces are sometimes well fitted with some sophis-ticated nonlinear estimates, e.g.:

Theorem 1.26 (Malek, Prazak, Steinhauer [269]57). If p ≥ 2 and ε > 0, thenthere is c > 0 such that

c∥∥u‖W 2/p−ε,p(Ω) ≤

(∥∥u∥∥pLp(Ω)

+

∫Ω

|u|p−2|∇u|2 dx)1/p

. (1.46)

1.3 Nemytskiı mappings

Considering integers j, m0, m1, . . . , mj , we say that a mapping a : Ω×Rm1 ×· · ·×Rmj → Rm0 is a Caratheodory mapping if a(·, r1, . . . , rj) : Ω → Rm0 is measurablefor all (r1, . . . , rj) ∈ Rm1 × · · · × Rmj and a(x, ·) : Rm1 × · · · × Rmj → Rm0

is continuous for a.a. x ∈ Ω. Then the so-called Nemytskiı mappings Na mapfunctions ui : Ω → Rmi , i = 1, . . . , j, to a function Na(u1, . . . , uj) : Ω → Rm0

defined by [Na(u1, . . . , uj)

](x) = a

(x, u1(x), . . . , uj(x)

). (1.47)

Theorem 1.27 (Nemytskiı mappings in Lebesgue spaces58). If a : Ω×Rm1 ×· · · × Rmj → Rm0 is a Caratheodory mapping and the functions ui : Ω → Rmi ,i = 1, . . . , j, are measurable, then Na(u1, . . . , uj) is measurable. Moreover, if asatisfies the growth condition

∣∣a(x, r1, . . . , rj)∣∣ ≤ γ(x) + C

j∑i=1

∣∣ri∣∣pi/p0for some γ∈Lp0(Ω), (1.48)

with 1 ≤ pi < +∞, 1 ≤ p0 < +∞, then Na is a bounded continuous mappingLp1(Ω;Rm1) × · · · × Lpj (Ω;Rmj ) → Lp0(Ω;Rm0). If some pi = +∞, i = 1, . . . , j,the same holds if the respective term |·|pi/p0 is replaced by any continuous function.

The case p0 = +∞ has to be excluded, cf. Exercise 2.61. Also, it should beemphasized that Na cannot be weakly continuous unless a(x, ·) is affine for a.a. x ∈Ω; cf. Exercise 2.62. A “canonical” example is a sequence uk(x) := sign(sin(kπx)),Ω = (0, 1) ⊂ R1, which converges weakly (or weakly*) to 0 in any Lp(0, 1), but fora(x, r) = |r| one obviously gets (cf. Figure 3) that w-lim

k→∞a(uk) = lim

k→∞1 = 1 �=

0 = a(0) = a(w-limk→∞

uk

).

57In fact, [269] works with periodic functions having zero means. The proof relies on theso-called Nikolskiı spaces. Here, we present an obvious modification involving the Lp-norm in(1.46).

58Cf. Exercise 2.60 below.


u

1/kuk |uk|

weak (here evenstrong) convergenceweak

convergenceNemystkiımapping

Nemystkiımapping

uk �→ |uk|

u �→ |u| 00 11

Fig. 3. A counterexample for weak continuity of nonlinear Nemytskiı mappings.

Sometimes, it is useful to consider Nemytskiı mappings in a Sobolev space.Here we confine ourselves to a special case of a one-argument Nemytskiı mappingwith an integrand independent of x; in fact, it is then a usual superposition.

Proposition 1.28 (Superposition operator in Sobolev spaces59). If a : R →R is Lipschitz continuous, then a(u) ∈ W 1,p(Ω) for any u ∈ W 1,p(Ω), p ∈ [1,+∞],and moreover it holds that

∇a(u) = a′(u)∇u (a.e. on Ω). (1.49)

Denoting u+ = max(u, 0) and u− = min(u, 0), Proposition 1.28 yields60

∇(u+) =

{∇u if u > 0,0 if u ≤ 0,

∇(u−) ={

0 if u ≥ 0,∇u if u < 0,

(a.e. on Ω). (1.50)

Another useful assertion addresses the change of the order of an integra-tion and a differentiation, and is based on the Lebesgue dominated convergenceTheorem 1.14.

Theorem 1.29 (Change of integration and differentiation). Let (x, r) �→ϕ : Ω×R → R and its partial derivative ∂

∂rϕ : Ω×R → R be Caratheodory

functions, |ϕ(·, 0)| ∈ L1(Ω) and the collection { ∂∂rϕ(·, r)}|r|≤ε have a common

integrable majorant. Then Φ : r �→ ∫Ω ϕ(x, r) dx is differentiable at 0 and

ddtΦ(0) =

∫Ω

∂∂rϕ(x, 0) dx.

1.4 Green formula and some inequalities

Let us define the unit outward normal ν = ν(x) ∈ Rn to the boundary Γ at apoint x ∈ Γ as the vector

ν(x) =

Ai

(∂gi∂ξ1

, . . . ,∂gi

∂ξn−1, 1

)√∣∣∣∂gi

∂ξ1

∣∣∣2 + · · ·+∣∣∣ ∂gi∂ξn−1

∣∣∣2 + 1

(1.51)

59See, e.g., monographs by Cazenave and Haraux [90, Prop. 1.3.5] or Ziemer [430, Theo-rem 2.1.11]. Original results are, in particular, due to Marcus and Mizel [273, 274].

60See, e.g., Cazenave and Haraux [90, Corollary 1.3.6] or Ziemer [430, Corollary 2.1.8].

1.4. Green formula and some inequalities 21

where x = Aiξ = Ai

(ξ1, . . . , ξn−1, gi(ξ1, . . . , ξn−1)

)as in (1.32). It is again natural

to assume Ω a Lipschitz domain. Then the functions gi are Lipschitz continuousand, by the so-called Rademacher theorem [342], they possess derivatives almosteverywhere on Γi ⊂ Rn−1, cf. (1.32), hence ν is defined almost everywhere onΓ and does not depend on the concrete covering of Γ by the coordinate systems(Gi, Ai). For a function f : Γ → R, we can define the surface integral

∫Γ f(x) dS

through (n− 1)-dimensional Lebesgue measure as∫Γ

f(x) dS =∑i

∫Gi

f(Ai(ξ1, . . . , ξn−1, gi(ξ1, . . . , ξn−1)

)×√∣∣∣∂gi

∂ξ1

∣∣∣2+ · · ·+∣∣∣ ∂gi∂ξn−1

∣∣∣2+ 1 d(ξ1, . . . , ξn−1) (1.52)

where Gi ⊂ Gi are chosen suitably to realize, instead of the overlapping coveringas on Figure 2, a disjoint covering of Γ. Again, the value

∫Γf(x) dS does not

depend on the particular coordinate systems (Gi, Ai).

Theorem 1.30 (Multidimensional by-part integration). If v∈W 1,p(Ω) andz∈W 1,p′

(Ω), then

∀i = 1, . . . , n :

∫Ω

(v∂z

∂xi+

∂v

∂xiz)dx =

∫Γ

vz νi dS. (1.53)

Considering z = (z1, . . . , zn), writing (1.53) for zi instead of z, and eventuallysumming it over i = 1, . . . , n with abbreviating div z :=

∑ni=1

∂∂xi

zi the divergenceof the vector field z, we arrive atthe formula which we will often use:

Theorem 1.31 (Green’s formula [188]61). For any v ∈ W 1,p(Ω) and z ∈W 1,p′

(Ω;Rn), the following formula holds:∫Ω

(v (div z) + z ·∇v

)dx =

∫Γ

v (z ·ν) dS. (1.54)

Theorem 1.32 (Poincare-type inequalities [339]). Let 1 ≤ q ≤ p∗. Thenthere is C

P< +∞ such that

‖u‖W 1,p(Ω) ≤ CP

(‖∇u‖Lp(Ω;Rn) + ‖u‖Lq(Ω)

). (1.55)

Let 1≤q≤p#, let Ω be connected 62, and let ΓD,ΓN ⊂ Γ be such that measn−1(ΓD) >0 and measn−1(ΓN) > 0. Then there is C

P< +∞ such that

‖u‖W 1,p(Ω) ≤ CP

(‖∇u‖Lp(Ω;Rn) +∥∥u|ΓN∥∥Lq(ΓN)

)(1.56)

61Putting z = ∇u into (1.54), we get∫Ω vΔu +∇v · ∇udx =

∫Γ v ∂

∂νu dS derived in [188]. In

fact, (1.54) holds, by continuous extension, even under weaker assumptions, cf. Necas [302].62This means that, for any x0, x1 ∈ Ω, there is z : [0, 1] → Ω continuous with z(0) = x0 and

z(1) = x1. It has the consequence that functions with zero gradient must be constant on Ω.


andu|ΓD = 0 ⇒ ‖u‖W 1,p(Ω) ≤ C

P‖∇u‖Lp(Ω;Rn), (1.57)

and also ∥∥u∥∥W 1,p(Ω)

≤ CP

(∥∥∇u∥∥Lp(Ω;Rn)

+∣∣∣ ∫

Ω

u dx∣∣∣). (1.58)

A special case of (1.56) with ΓD = Γ and p = q = 2, is sometimes also calledFriedrichs’ inequality [157].

Theorem 1.33 (Korn inequality [237]63). Let 1 < p < +∞. There is a constantC

K= C

K(p,Ω) such that for any v ∈ W 1,p

0 (Ω) it holds that

∥∥v∥∥W 1,p

0 (Ω;Rn)≤ C

K

∥∥e(v)∥∥Lp(Ω;Rn×n)

where e(v) =∇v + (∇v)

2. (1.59)

1.5 Bochner spaces

We will now define spaces of abstract functions on a bounded interval I ⊂ Rvalued in a Banach space V , invented by Bochner [59], which is a basic tool onthe abstract level for Part II. We say that u : I → V is simple if it takes onlyfinite number of values vi ∈ V and Ai := u−1(vi) is Lebesgue measurable; then∫ T

0 u(t) dt :=∑

finitemeas1(Ai)vi. We say that u : I → V is Bochner measurableif it is a point-wise limit (in the strong topology) of V of a sequence {uk}k∈Nof simple functions; i.e. uk(t) → u(t) for a.a. t ∈ I. We say that u : I → V is

absolutely continuous64 if, for each ε > 0, there is δ > 0 such that∑K

k=1 ‖u(tk)−u(sk)‖V ≤ ε whenever tk−1 ≤ sk ≤ tk ≤ T for k = 1, . . . ,K ∈ N, t0 = 0,

and∑K

k=1 tk − sk ≤ δ. A point t ∈ I is called a Lebesgue point of u : I → V if

limh↘01h

∫ h/2

−h/2‖u(t+ϑ) − u(t)‖dϑ = 0. Analogously, a right Lebesgue point t∈ I

means limh↘01h

∫ h

0‖u(t+ϑ)− u(t)‖dϑ = 0.

Theorem 1.34 (Pettis [334]65). If V is separable, then u is Bochner measurableif and only if it is weakly measurable in the sense: 〈v∗, u(·)〉 is Lebesgue measurablefor any v∗ ∈ V ∗.

Considering simple functions {uk}k∈N as above, we call u : I → V

Bochner integrable if limk→∞∫ T

0‖u(t) − uk(t)‖V dt = 0. Then

∫ T

0u(t) dt :=

limk→∞∫ T

0uk(t) dt; this limit exists and is independent of the particular choice

63See Necas [307] or monographs [130, 308, 427]. A counterexample for p=1 is by Ornstein[318].

64If V = R1, this definition naturally coincides with the absolute-continuity with respect tothe Lebesgue measure on I used on p. 12.

65In fact, [334] works with a general Banach space, showing equivalence of the Bochner mea-surable mappings with a.e. separably valued weakly measurable ones. See Example 1.42 for aweakly* continuous function u valued in V = L∞(0, 1) which is not Bochner measurable.

1.5. Bochner spaces 23

of the sequence {uk}k∈N. Moreover, if V is separable, then a Bochner measur-able function u is Bochner integrable if and only if ‖u(·)‖V is Lebesgue inte-

grable. Then also∥∥ ∫ T

0u(t) dt‖V ≤ ∫ T

0‖u(t)‖V dt. From this, we can see that

limh↘01h

∫ h/2

−h/2u(t+ϑ)dϑ → u(t) at each Lebesgue point t∈I.66

Theorem 1.35. If u∈L1(I;V ), then a.e. t∈I is a Lebesgue point for u.

An analogous assertion holds for right Lebesgue points.

An example for Bochner integrable functions is any u ∈ C(I; (V,weak)).67

However, some classical “scalar-valued” results need not hold in vectorial cases68.

For 1 ≤ p < ∞, a Bochner space Lp(I;V ) is the linear space (of classes withrespect to equivalence a.e.) of Bochner integrable functions u : I → V satisfying∫ T

0‖u(t)‖pV dt < +∞. This space is a Banach space if endowed with the norm

‖u‖Lp(I;V ) :=

(∫ T

0

∥∥u(t)∥∥pVdt

)1/p

. (1.60)

For p = ∞, we modify it standardly, i.e. ‖u‖L∞(I;V ) := ess supt∈I‖u(t)‖V . Later,we will often use partition of I to subintervals of the length τ := 2−KT , K ∈ N.

Proposition 1.36 (Density of piecewise constant functions). If 1 ≤ p <+∞, then the set {v : I → V ; ∃K ∈ N : ∀1 ≤ k ≤ 2K : v|((k−1)τ,kτ) is constant,

τ = 2−kT } is dense in Lp(I;V ). In particular, if p ∈ [1,+∞) and V is separable,Lp(I;V ) is separable too.

Proposition 1.37 (Uniform convexity). If V is uniformly convex and 1 < p <+∞, then Lp(I;V ) is uniformly convex, too.

Proposition 1.38 (Dual space). If p ∈ [1,+∞), the dual space to Lp(I;V ) alwayscontains Lp′

(I;V ∗) and the equality holds if V ∗ is separable, the duality pairingbeing given by the formula

⟨f, u

⟩Lp′(I;V ∗)×Lp(I;V )

:=

∫ T

0

⟨f(t), u(t)

⟩V ∗×V

dt . (1.61)

Thus, if p ∈ (1,+∞) and V is reflexive and separable, then Lp(I;V ) is reflexive.

66This follows simply from ‖u(t) − 1h

∫ h/2−h/2

u(t+ϑ)dϑ‖V = ‖ 1h

∫ h/2−h/2

u(t+ϑ) − u(t)dϑ‖V ≤1h

∫ h/2−h/2

‖u(t+ϑ)− u(t)‖V dϑ.67The weak* continuity would not be sufficient, however, as Example 1.42 shows.68E.g., u ∈ C(I; (V,weak∗)) need not be Bochner integrable, as Example 1.42 shows. Also, a

σ-additive absolutely continuous mapping u : I → V need not be represented by an integrablefunction v in the sense u(t) =

∫ t0 v(ϑ)dϑ; for the counterexample see Yosida [425, Sect.V.5].


Theorem 1.39 (Komura [235]). If V is reflexive and u : I → V is absolutely

continuous, then u is (strongly) differentiable a.e.69 and u(t) = u(0) +∫ t

0 u′, u′ ∈

L1(I;V ).70

Convention 1.40. Often, we will use V = Lq(Ω;Rm). Considering p, q ∈ [1,+∞],it is then natural to identify Lp(I;Lq(Ω;Rm)) with{

u : I × Ω → Rm;

∫ T

0

(∫Ω

∣∣u(t, x)∣∣qdx)p/qdt < +∞}

(1.62)

through the natural isomorphism u �→ u: [u(t)](x) := u(t, x). Likewise,Lp(I;W 1,q(Ω)) is identified via this isomorphism with functions u : I ×Ω → R for

which∫ T

0

(∫Ω |u(t, x)|q + |∇u(t, x)|qdx)p/q dt < +∞.

Proposition 1.41 (Interpolation of Lp1(I;Lq1(Ω)) and Lp2(I;Lq2(Ω)) 71). Letp1, p2, q1, q2∈ [1,+∞], λ∈ [0, 1], and v∈Lp1(I;Lq1(Ω)) ∩ Lp2(I;Lq2(Ω)). Then

1

p=

λ

p1+

1−λ

p2and

1

q=

λ

q1+

1−λ

q2

=⇒ ∥∥v∥∥Lp(I;Lq(Ω))

≤ ∥∥v∥∥λLp1(I;Lq1(Ω))

∥∥v∥∥1−λ

Lp2(I;Lq2(Ω)). (1.63)

Example 1.42. (The space L∞(I;L∞(Ω)) is not L∞(Q).72) Using the isomorphismfrom Convention 1.40, we can identify abstract functions from L∞(I;L∞(Ω)) withfunctions on Q := I × Ω. However, L∞(I;L∞(Ω)) �= L∞(Q). For Ω = I, thefunction u(t) = χ[0,t] induces a function u(t, x) = 1 if x ≤ t and = 0 if x > t whichobviously belongs to L∞(Q). Even u is weakly* continuous but it is not Bochnermeasurable, hence u �∈ L∞(I;L∞(Ω)).

Considering Banach spaces V0, V1, . . . , Vk and a : I × V1 × · · · × Vk → V0,let us still define the Nemytskiı mappings Na again by the formula (1.47). Thefollowing generalization of Theorem 1.27 holds:

Theorem 1.43 (Nemytskiı mappings in Bochner spaces [265]73). Let V0, V1,. . . , Vk be separable Banach spaces, a : I × V1 × · · · × Vk → V0 a Caratheodorymapping74 and the growth condition

∥∥a(t, r1, . . . , rk)∥∥V0≤ γ(t) + C

k∑i=1

∥∥ri∥∥pi/p0

Vifor some γ∈Lp0(I), (1.64)

69This means limε→01εu(t+ ε)− 1

εu(t) does exist for a.a. t ∈ I. The reflexivity of X is indeed

necessary, e.g. u : I → X := L∞(0, 1) defined by [u(t)](x) := x sin(t/x) is Lipschitz continuousbut nowhere differentiable.

70If V is a uniformly convex space, this formula has been proved already by Clarkson [97].71Cf. Exercise 8.60 on p.266.72This observation is due to Fattorini [144, Example 5.0.10].73Besides the original paper [265] by Lucchetti and Patrone, we refer also, e.g., to Hu and

Papageorgiou [209, Part I, Sect.9.1].74Here, it means that, for a.a. t ∈ I, a(t, ·) : V1×· · ·×Vk → V0 is to be (norm,norm)-continuous

and, as in Section 1.3, a(·, r1, . . . , rk) : I → V0 is to be Bochner measurable.

1.6. Some ordinary differential equations 25

hold with p0, p1, . . . , pk as for (1.48). Then Na maps Lp1(I;V1)× · · · ×Lpk(I;Vk)continuously into Lp0(I;V0).

1.6 Some ordinary differential equations

As an auxiliary problem, we will often use the initial-value problem for the systemof k ordinary differential equations in the form:

du

dt= f

(t, u(t)

)for a.a. t∈I, u(0) = u0, (1.65)

where f : (0,+∞)×Rk → Rk is a Caratheodory mapping. By a solution on a timeinterval [0, T ] we will understand an absolutely continuous mapping u : [0, T ] → Rk

such that the equation (1.65) holds a.e. on [0, T ] and u(0) = u0 holds, too.

Theorem 1.44 (Local-in-time existence). Let f : (0,+∞) × Rk → Rk be aCaratheodory mapping. Then there is T > 0 (not given a-priori) such that theinitial-value problem has a solution on the interval [0, T ].

The main ingredient for estimation of evolution systems in general is theso-called Gronwall inequality,75 which we will also often use. In the general form,this inequality says that, for all t ≥ 0,

y(t) ≤(C +

∫ t

0

b(θ)e−∫ θ0a(ϑ)dϑdθ

)e∫ t0a(θ)dθ (1.66)

whenever we know that y(t) ≤ C +∫ t

0(a(ϑ)y(ϑ) + b(ϑ))dϑ for some a, b ≥ 0 inte-

grable.76 For a ≥ 0 constant, (1.66) simplifies to y(t) ≤ eat(C +∫ t

0b(ϑ)e−aϑdϑ) ≤

(C +∫ t

0b(ϑ)dϑ)eaT for t ∈ I.

Theorem 1.45 (Existence and uniqueness). Let T be fixed and f : I×Rk → Rk

be a Caratheodory mapping satisfying the growth condition |f(t, r)| ≤ γ(t) + C|r|with some γ ∈ L1(I). Then:(i) The initial-value problem (1.65) has a solution u ∈ W 1,1(I;Rk) on the interval

I = [0, T ].

(ii) If f(t, ·) is also Lipschitz continuous in the sense |f(t, r1)−f(t, r2)| ≤ (t)|r1−r2| with some ∈ L1(I), then the solution is unique.

75In the general form presented here, which can be found, e.g., in Ioffe and Tihomirov [211,Sect. 9.1, Lemma 3], it is also called the Bellman-Gronwall inequality according to the originalworks [193] (for C = 0, a, b constant) and [41] (for C ≥ 0, b = 0, a ∈ L1(0, T )).

76A generalization for slightly superlinear growth in y, namely y ln(y) or y ln(y) ln(ln(y)), ispossible, too.


Applying the so-called bootstrap argument, i.e. knowing that u ∈ W 1,1(I;Rk)and therefore f(t, u) ∈ W 1,1(I;Rk) provided f is smooth in the sense

∀ρ∈R+ ∃γρ∈L1(I), Cρ < +∞ ∀t∈I, |r| ≤ ρ :∣∣∣∂f∂t

(t, r)∣∣∣ ≤ γρ(t) and |f(t, r)| ≤ Cρ, (1.67)

we know immediately that ddtu = f(t, u) ∈ W 1,1(I;Rk).77 Hence:

Theorem 1.46 (Regularity). If f satisfies all assumptions in Theorem 1.45 andis also smooth in the sense (1.67), then u ∈ W 2,1(I;Rk).

The following discrete version of the Gronwall inequality will be often used:78

yl ≤(C + τ

l−1∑k=1

bk

)eτ

∑l−1k=1 ak (1.68)

provided yl ≤ C + τ∑l−1

k=1(akyk + bk) for any l ≥ 0 (of course, for l = 0 it meansyl ≤ C). We will often use ak ≡ a constant, and the condition

yl ≤ C + τ

l∑k=1

(ayk + bk

), (1.69)

from which we can easily derive yl ≤ (1− aτ)−1(C + τb1 + τ∑l−1

k=1(ayk + bk+1)),so that (1.68) gives

yl ≤ eτla/(1−aτ)

1− aτ

(C + τ

l∑k=1

bk

)if τ <

1

a. (1.70)

There is a variant of the discrete Gronwall inequality by Nochetto, Savareand Verdi [314] designed for fine estimates for variable time steps and even bet-ter fit to discretisation of some evolutionary problems. While the condition (1.69)is an analog of d

dty ≤ ay + b, the modified version is based rather on the con-

tinuous condition79 y ddty ≤ ay+b+cy2: If non-negative sequences {yk}Kk=0 and

{(zk, ak, bk)}Kk=1 satisfy80

ykyk − yk−1

τ+ zk ≤ akyk + bk + cy2k for k = 1, . . . ,K (1.71a)

77We use ddt

f(t, u) = ∂∂t

f(t, u)+ ∂∂r

f(t, u) ddt

u ∈ L1(I;Rk) because ∂∂t

f(t, u) ∈ L1(I;Rk) while

| ∂∂r

f(t, u)| ≤ � and ddt

u = f(t, u) ∈ L∞(I;Rk).78See Lees [255], Krejcı and Roche [241, Lemma A.1], Quarteroni and Valli [341, Lemma 1.4.2]

or Thomee [405].79Note that 1

2ddt

y2 = y ddt

y ≤ ay+ b+ cy2 ≤ ( 12a+ c)y2 + 1

2a+ b, from which the boundedness

of y follows by the classical Gronwall inequality (1.66) provided a, b, and c are integrable.80More specifically, this is (up to some scaling) a special case of [314, Lemma 4.12] for c

independent of k.

1.6. Some ordinary differential equations 27

with a constant c ≥ 0, then one has both maxk=1,...,K e−kcτ/(1−cτ)yk and

(2τ∑K

k=1 e−((2k−1)cτ)/(1−cτ)zk)

1/2 estimated from above by

(y20 + 2τ

K∑k=1

e−(2k−1)cτ bk)1/2+

√2τ

K∑k=1

e−(k−1)cτak

provided 0 < τ < 1/c. A particular but well applicable case can be obtained forK = T/τ : assuming (1.71a) and also

0 < τ ≤ τ0 with τ0 <1

c, (1.71b)

one has, with some constant CT independent of τ ,

maxk=1,...,K

yk +(τ

T/τ∑k=1

zk

)1/2≤ CT

(y0 +

(τ

T/τ∑k=1

bk

)1/2+ τ

T/τ∑k=1

ak

). (1.72)

Part I

STEADY-STATE PROBLEMS

Chapter 2

Pseudomonotone or weaklycontinuous mappings

The basic modern approach to boundary-value problems in differential equationsof the type (0.1)–(0.2) is the so-called energy-method technique which took thename after a-priori estimates having sometimes physical analogies as bounds of anenergy.1 This technique originated from modern theory of linear partial differentialequations where, however, other approaches are efficient, too. On the abstractlevel, this method relies on relative weak compactness of bounded sets in reflexiveBanach spaces, and either pseudomonotonicity or weak continuity of differentialoperators which are understood as bounded from one Banach space to another(necessarily different) Banach space. On the concrete-problem level, the main toolis a weak formulation of boundary-value problems in question, Poincare and Holderinequalities, and fine issues from the theory of Sobolev spaces.

2.1 Abstract theory, basic definitions, Galerkin method

Throughout this chapter (and most of the others), V will be a separable reflexiveBanach space and V ∗ its dual space, with ‖ · ‖ and ‖ · ‖∗ denoting briefly theirnorms instead of ‖ · ‖V and ‖ · ‖V ∗ , respectively.

Definition 2.1 (Monotonicity modes). Let A be a mapping V → V ∗.

(i) A : V → V ∗ is monotone iff ∀u, v ∈ V : 〈A(u)−A(v), u − v〉 ≥ 0.

(ii) If A is monotone and u �= v implies 〈A(u)−A(v), u−v〉 > 0, then A is strictlymonotone.

1Cf. Example 6.7 or e.g. also (11.120) or (12.11).


32 Chapter 2. Pseudomonotone or weakly continuous mappings

(iii) Considering an increasing function d : R+ → R, we say that A : V → V ∗ isd-monotone with respect to a seminorm | · |,

⟨A(u)−A(v), u − v

⟩ ≥ (d(|u|)− d(|v|))(|u| − |v|). (2.1)

If | · | is the norm ‖ · ‖ on V , we say simply that A is d-monotone. Moreover,A is called uniformly monotone if⟨

A(u)−A(v), u − v⟩ ≥ ζ

(‖u− v‖)‖u− v‖ (2.2)

for some increasing continuous function ζ : R+ → R+. If ζ(r) = δr for someδ > 0, then A is called strongly monotone.

(iv) The mapping A : V → V ∗ is called pseudomonotone iff

A is bounded, and (2.3a)

uk ⇀ ulim supk→∞

⟨A(uk), uk−u

⟩ ≤ 0

}⇒ ∀v∈V :⟨

A(u), u−v⟩ ≤ lim inf

k→∞⟨A(uk), uk−v

⟩. (2.3b)

Remark 2.2. Let us emphasize that the monotonicity due to Definition 2.1(i) hasno direct relation with monotonicity of mappings with respect to an ordering. E.g.,if V ∗ = V , the composition of monotone operators has a good sense but need notbe monotone. Definition 2.1(iv) represents a suitable extent2 of generalization ofthe monotonicity concept from the viewpoint of quasilinear differential equationsof the type (0.2).

Definition 2.3 (Continuity modes).(i) A : V→V ∗ is hemicontinuous iff ∀u, v, w∈V the function t �→ 〈A(u+tv), w〉 is

continuous, i.e. A is directionally weakly continuous.

(ii) If it holds only for v = w, i.e. ∀u, v∈V : t �→ 〈A(u+tv), v〉 is continuous, thenA is called radially continuous.

(iii) A : V→V ∗ is demicontinuous iff ∀w∈V the functional u �→ 〈A(u), w〉 is con-tinuous; i.e. A is continuous as a mapping (V, norm) → (V ∗,weak).

(iv) A : V→V ∗ is weakly continuous iff ∀w∈V the functional u �→ 〈A(u), w〉 isweakly continuous; i.e. A is continuous as a mapping (V,weak) → (V ∗,weak).

(v) A : V→V ∗ is totally continuous if it is continuous as a mapping (V,weak) →(V ∗, norm).

Lemma 2.4. Any pseudomonotone mapping A is demicontinuous.

Proof. Suppose uk → u. By (2.3a), the sequence {A(uk)}k∈N is bounded in areflexive space V ∗. Then, as V is assumed separable, by the Banach Theorem 1.7after taking a subsequence (denoted, for simplicity, by the same indices) we have

2In the sense that the premise of (2.3b) still can be proved under reasonable assumptions andthe conclusion of (2.3b) still suffices to prove convergence of various approximate solutions.

2.1. Abstract theory, basic definitions, Galerkin method 33

A(uk) ⇀ f for some f ∈ V ∗. Then limk→∞〈A(uk), uk − u〉 = 〈f, u − u〉 = 0 andtherefore, by (2.3b),⟨

A(u), u− v⟩ ≤ lim inf

k→∞⟨A(uk), uk − v

⟩= 〈f, u− v〉 (2.4)

for any v ∈ V . From this we get A(u) = f . In particular, f is determined uniquely,and thus even the whole sequence (not only the selected subsequence) must con-verge. �

Definition 2.5 (Coercivity). A : V → V ∗ is coercive iff ∃ζ : R+ → R+:lims→+∞ ζ(s) = +∞ and 〈A(u), u〉 ≥ ζ(‖u‖)‖u‖. In other words, A coercive means

lim‖u‖→∞

⟨A(u), u

⟩‖u‖ = +∞. (2.5)

Theorem 2.6 (Brezis [64]). Any A pseudomonotone and coercive is surjective;this means, for any f ∈ V ∗, there is at least one solution to the equation

A(u) = f. (2.6)

Proof. Let us divide the proof into four particular steps.

Step 1: (Abstract Galerkin approximation.) As V is supposed separable, we cantake a sequence of finite-dimensional subspaces

∀k ∈ N : Vk ⊂ Vk+1 ⊂ V and⋃k∈N

Vk is dense in V . (2.7)

Then we define a Galerkin approximation uk ∈ Vk by the identity:

∀v ∈ Vk :⟨A(uk), v

⟩= 〈f, v〉. (2.8)

Step 2: (Existence of approximate solutions uk.) In other words, we seek uk ∈ Vk

solving I∗k (A(uk) − f) = 0 where Ik : Vk → V is the canonical inclusion so thatthe adjoint operator I∗k : V ∗ → V ∗k represents the restriction I∗kf = f |Vk

. Besides,as Vk is finite-dimensional, we will identify Vk

∼= V ∗k by a linear homeomorphismJk : Vk → V ∗k such that 〈Jku, u〉 = ‖u‖2Vk

, ‖Jku‖V ∗k= ‖u‖Vk

, and 〈Jku, J−1k f〉 =

〈f, u〉.3As A is coercive, for � sufficiently large we have

‖u‖Vk= � =⇒ ⟨

A(u)− f, u⟩ ≥ ⟨A(u), u⟩− ‖f‖∗‖u‖ > 0. (2.9)

3If necessary, we can re-norm the finite-dimensional Vk to impose a Hilbert structure (i.e. Vk

is then homeomorphic with a Euclidean space). Then Jk can be taken as in (3.1) below; notethat, by Lemma 3.2, Jk is a homeomorphism. Also note that (2.5) restricted on Vk holds inthe new, equivalent norm, as well; possibly, the function ζ in Definition 2.5 is changed by thisrenormalization.


Suppose, for a moment, that I∗kA(u) �= I∗kf for any u ∈ Vk with ‖u‖Vk≤ �. Then

the mapping

u �→ �J−1k I∗k

(f −A(u)

)∥∥I∗k (f −A(u))∥∥V ∗k

(2.10)

maps the convex compact set {u ∈ Vk; ‖u‖ ≤ �} into itself because ‖J−1k ‖ = 1;

note that ‖J−1k f‖Vk

= ‖f‖V ∗k. Also, by Lemma 2.4, the mapping u �→ 〈A(u), v〉 :

Vk → R is continuous for any v so that also u �→ I∗kA(u) : Vk → V ∗k is continuous.By the Brouwer fixed-point Theorem 1.10, the mapping (2.10) has a fixed pointu, this means

u = �J−1k I∗k

(f −A(u)

)∥∥I∗k (f −A(u))∥∥V ∗k

. (2.11)

As ‖J−1k f‖Vk

= ‖f‖V ∗k, (2.11) implies ‖u‖Vk

= �. Testing (2.11) by Jku‖I∗k (f −A(u))‖V ∗

k, one gets

�2∥∥I∗k (f −A(u))

∥∥V ∗k

= 〈Jku, u〉∥∥I∗k (f −A(u))

∥∥V ∗k

= �⟨Jku, J

−1k I∗k (f −A(u))

⟩= �

⟨I∗k (f −A(u)), u

⟩= �

⟨f −A(u), Iku

⟩= �

⟨f −A(u), u

⟩(2.12)

which yields 〈A(u)− f, u〉 = −�‖I∗k(A(u)− f)‖V ∗k≤ 0, a contradiction with (2.9).

Step 3: (An a-priori estimate.) Moreover, putting v := uk into (2.8), we canestimate4

ζ(‖uk‖

)‖uk‖ ≤ ⟨A(uk), uk

⟩= 〈f, uk〉 ≤ ‖f‖∗‖uk‖ (2.13)

with a suitable increasing function ζ : R+ → R+ such that limξ→∞ ζ(ξ) = +∞, cf.the coercivity (2.5) of A. Then ‖uk‖ ≤ ζ−1(‖f‖∗) < +∞, so that uk is boundedin V independently of k. This holds even for any solution to (2.8).

Step 4: (Limit passage.) Since {uk}k∈N is bounded and V is reflexive and sepa-rable, by the Banach Theorem 1.7 together with Proposition 1.3, there is a sub-sequence and u ∈ V such that uk ⇀ u. From (2.8), we have also⟨

A(uk), vm − uk

⟩= 〈f, vm − uk〉 (2.14)

for any k ≥ m and vm ∈ Vm ⊂ Vk. By density of⋃

k∈N Vk in V , we can takevk → u. Then, by (2.14), one gets

lim supk→∞

〈A(uk), uk − u〉 = lim supk→∞

(〈A(uk), uk − vk〉+ 〈A(uk), vk − u〉

)≤ lim

k→∞

(〈f, uk − vk〉+ ‖A(uk)‖∗‖vk − u‖

)= 0. (2.15)

4Here we forget possible renormalization of the finite-dimensional subspaces Vk and comeback to the original norm on V .

2.2. Some facts about pseudomonotone mappings 35

Note that the sequence {uk}k∈N has been proved bounded so {‖A(uk)‖∗}k∈N isbounded by (2.3a) and that, in fact, even an equality holds in (2.15) and “limsup”is a limit. By pseudomonotonicity (2.3b) of A, we get

∀v∈V : lim infk→∞

⟨A(uk), uk − v

⟩ ≥ ⟨A(u), u− v⟩. (2.16)

On the other hand, from (2.14) we also have

∀v∈⋃m∈N

Vm : lim infk→∞

⟨A(uk), uk − v

⟩= lim

k→∞〈f, uk − v〉 = 〈f, u− v〉. (2.17)

Combining (2.16) and (2.17), one gets 〈A(u), u− v〉 ≤ 〈f, u− v〉 for any v rangingover a dense subset of V , namely

⋃m∈N Vm, which shows that A(u) = f . �

Remark 2.7 (Nonconstructivity). Let us emphasize three aspects of high noncon-structivity of the above proof:� usage of Brouwer’s fixed-point theorem,� a contradiction argument, and� a selection of a convergent subsequence by a compactness argument.

Remark 2.8 (Necessity of approximation). The approximation (Step 1) is necessaryin the above proof, otherwise one would have to think about usage of Schauder’stype fixed point Theorem 1.9 instead of the Brouwer one. This would need ad-ditional assumptions about weak continuity of A and the Hilbert structure of V ,cf. Exercise 2.56, which is not fitted with general quasilinear differential equations,cf. Sect. 2.5 where omitting the approximation would also hurt for not allowingfor a weaker concept of A as V → Z∗ with Z � V .

2.2 Some facts about pseudomonotone mappings

Brezis’ Theorem 2.6 showed the importance of the class of pseudomonotone map-pings. It is therefore worth knowing some more specific cases leading to suchmappings.

Lemma 2.9 (Brezis [64]). Radially continuous monotone mappings satisfy (2.3b).In particular, bounded radially continuous monotone mappings are pseudomono-tone.

Proof. Take uk ⇀ u and assume lim supk→∞〈A(uk), uk − u〉 ≤ 0. Since A ismonotone, 〈A(uk), uk − u〉 ≥ 〈A(u), uk − u〉 → 0 so that lim infk→∞〈A(uk), uk −u〉 ≥ 0 and therefore altogether

limk→∞

〈A(uk), uk − u〉 = 0. (2.18)

We take uε = (1−ε)u + εv, ε > 0, and write the monotonicity condition of Abetween uk and uε:

0 ≤ 〈A(uk)−A(uε), uk − uε〉 = 〈A(uk)−A(uε), ε(u − v) + uk − u〉 (2.19)


and, by a simple algebraic manipulation, we obtain

ε〈A(uk), u− v〉 ≥ 〈A(uε), uk − u〉 − 〈A(uk), uk − u〉+ ε〈A(uε), u− v〉. (2.20)

Therefore, fixing ε > 0 and passing with k to infinity, by (2.18) we get

ε lim infk→∞

〈A(uk), u− v〉 ≥ ε〈A(uε), u− v〉. (2.21)

Then divide it by ε, which gives lim infk→∞〈A(uk), u − v〉 ≥ 〈A(uε), u − v〉 =〈A(u + ε(v−u)), u−v〉. Passing with ε → 0 and using the radial continuity of A,we eventually get

lim infk→∞

〈A(uk), u− v〉 ≥ limε↘0

〈A(u + ε(v−u)), u−v〉 = 〈A(u), u−v〉. (2.22)

The pseudomonotonicity of A then follows by using (2.22) with (2.18):

lim infk→∞

〈A(uk), uk−v〉 = limk→∞

〈A(uk), uk−u〉+ lim infk→∞

〈A(uk), u−v〉 ≥ 〈A(u), u−v〉.

�Lemma 2.10. Any bounded demicontinuous mapping A : V → V ∗ satisfying(

uk ⇀ u & lim supk→∞

⟨A(uk)−A(u), uk−u

⟩ ≤ 0)

⇒ uk → u (2.23)

is pseudomonotone.

Proof. The premise of (2.3b), i.e. uk⇀u and lim supk→∞

〈A(uk), uk−u〉 ≤ 0, yields

lim supk→∞

⟨A(uk)−A(u), uk−u

⟩= lim sup

k→∞

⟨A(uk), uk−u

⟩− limk→∞

⟨A(u), uk−u

⟩ ≤ 0,

so that by (2.23) we have uk → u, and by demicontinuity of A also A(uk) ⇀ A(u),and eventually limk→∞〈A(uk), uk − v〉 = 〈A(u), u− v〉 for any v ∈ V . �Lemma 2.11.(i) The sum of any pseudomonotone mappings remains pseudomonotone, i.e. A1

and A2 pseudomonotone implies u �→ A1(u) +A2(u) pseudomonotone.

(ii) A shift of a pseudomonotone mapping remains pseudomonotone, i.e. A pseu-domonotone implies u �→ A(u + w) pseudomonotone for any w ∈ V .

Proof. The boundedness (2.3a) of A1 +A2 and A(·+w) is obvious hence we needto show only (2.3b).

To prove (i), let A1, A2 be pseudomonotone, uk ⇀ u and lim supk→∞〈[A1 +A2](uk), uk − u〉 ≤ 0. Let us verify that

lim supk→∞

⟨A1(uk), uk − u

⟩ ≤ 0 and lim supk→∞

⟨A2(uk), uk − u

⟩ ≤ 0. (2.24)

2.3. Equations with monotone mappings 37

Suppose, for a moment, that lim supk→∞〈A2(uk), uk − u〉 = ε > 0. Taking asubsequence, we can suppose that limk→∞〈A2(uk), uk − u〉 = ε > 0 and therefore

lim supk→∞

⟨A1(uk), uk − u

⟩ ≤ −ε < 0. (2.25)

As A1 is pseudomonotone, we get lim infk→∞〈A1(uk), uk − v〉 ≥ 〈A1(u), u− v〉 forany v ∈ V . In particular, for v = u we get lim infk→∞〈A1(uk), uk − u〉 ≥ 0, whichcontradicts (2.25). Thus (2.24) holds. By the pseudomonotonicity both for A1 andfor A2, we get

lim infk→∞

⟨[A1+A2](uk), uk−v

⟩ ≥ lim infk→∞

⟨A1(uk), uk−v

⟩+ lim inf

k→∞⟨A2(uk), uk−v

⟩≥ ⟨A1(u), u−v

⟩+⟨A2(u), u−v

⟩≥ ⟨[A1+A2](u), u−v⟩.

As to (ii), let uk ⇀ u and lim supk→0〈A(uk+w), uk−u〉 ≤ 0. Thenobviously uk+w ⇀ u+w and lim supk→0〈A(uk+w), (uk+w) − (u+w)〉 ≤ 0. IfA is pseudomonotone, then lim infk→0〈A(uk+w), uk−v〉 = lim infk→0〈A(uk+w),(uk+w)− (v+w)〉 ≥ 〈A(u+w), (u+w)− (v+w)〉 = 〈A(u+w), u−v〉, hence A(·+w)is pseudomonotone. �Corollary 2.12. A perturbation of a pseudomonotone mapping by a totally contin-uous mapping is pseudomonotone.

Proof. Realize that any totally continuous mapping is pseudomonotone; indeed,it is bounded (which can be easily proved by contradiction) and, if uk ⇀ u, thenA(uk) → A(u) and thus limk→∞〈A(uk), uk − v〉 = 〈A(u), u − v〉 so that (2.3b) istrivial. �

2.3 Equations with monotone mappings

Monotone mappings (with boundedness and radial continuity properties) are aspecial class of pseudomonotone mappings, cf. Lemma 2.9, and, as such, theyallow special treatment with a bit stronger results than a general “pseudomonotonetheory” can yield, cf. Theorem 2.14 vs. Proposition 2.17.

Lemma 2.13 (Minty’s trick [286]). Let A : V → V ∗ be radially continuous andlet 〈f−A(v), u−v〉 ≥ 0 for any v∈V . Then f = A(u).

Proof. Replace v with u+ εw with w ∈ V arbitrary. This gives⟨f −A(u+εw),−εw

⟩ ≥ 0. (2.26)

Divide it by ε > 0 and pass to the limit with ε by using radial continuity of A:

0 ≥ ⟨f −A(u+εw), w⟩→ ⟨

f −A(u), w⟩. (2.27)

As w is arbitrary, one gets A(u) = f . �


Theorem 2.14. Let A be bounded,5 radially continuous, monotone, coercive. Then:

(i) A is surjective; this means, for any f ∈ V ∗, there is u solving (2.6). Moreover,the set of solutions to (2.6) is closed and convex.

(ii) If, in addition, A is strictly monotone, then A−1 : V ∗ → V does exist, isstrictly monotone, bounded, and demicontinuous. If A is also d-monotone andV uniformly convex, then A−1 : V ∗ → V is continuous.

(iii) If, in addition, A is uniformly (resp. strongly) monotone, then A−1 : V ∗ → Vis uniformly (resp. Lipschitz) continuous.

Proof. By Lemma 2.9, A is pseudomonotone. As A is supposed also coercive, thesurjectivity of A follows from Theorem 2.6. By Lemma 2.4, A is demicontinuous,hence the set of solutions to (2.6) is closed in the norm topology of V . Hence, toprove convexity of this set, it suffices to show that u = 1

2u1 + 12u2 solves (2.6)

provided u1 and u2 do so, cf. Proposition 1.6. Thus we have⟨f−A(v), u−v

⟩=

1

2

⟨f −A(v), u1 − v

⟩+

1

2

⟨f −A(v), u2 − v

⟩=

1

2

⟨A(u1)−A(v), u1−v

⟩+

1

2

⟨A(u2)−A(v), u2−v

⟩ ≥ 0 (2.28)

because of A(u1) = f = A(u2) and of monotonicity of A. Then, by Lemma 2.13,one gets A(u) = f .

Let us go on to (ii). If A is strictly monotone, we have 〈A(u1)−A(u2), u1 −u2〉 = 〈f − f, u1 − u2〉 = 0 which is possible only if u1 = u2. In other words, theequation (2.6) has a unique solution so that the inverse A−1 does exist.

The mapping A−1 is strictly monotone: For f1, f2 ∈ V ∗, f1 �= f2, put ui =A−1(fi). Then also u1 �= u2. As A is strictly monotone, one has⟨

f1 − f2, A−1(f1)−A−1(f2)

⟩=⟨A(u1)−A(u2), u1 − u2

⟩> 0. (2.29)

The mapping A−1 is bounded: by the coercivity of A, there is ζ : R+ → Rsuch that limξ→∞ ζ(ξ) = +∞ and 〈A(u), u〉 ≥ ‖u‖ζ(‖u‖). Therefore

ζ(‖u‖)‖u‖ ≤ ⟨A(u), u⟩ = ⟨f, u⟩ ≤ ‖f‖∗‖u‖ (2.30)

so that ζ(‖A−1(f)‖) = ζ(‖u‖) ≤ ‖f‖∗. Thus A−1 maps bounded sets in V ∗ intobounded sets in V .

The mapping A−1 is demicontinuous, i.e. (norm,weak)-continuous: take fk →f in V ∗. As A−1 was shown to be bounded, {A−1(fk)}k∈N is bounded and (possiblyup to a subsequence) uk = A−1(fk) ⇀ u in V by Banach’s Theorem 1.7. It remainsto show A(u) = f . By the monotonicity of A, for any v ∈ V :

0 ≤ ⟨A(uk)−A(v), uk − v⟩=⟨fk −A(v), uk − v

⟩. (2.31)

5If proved directly, i.e. without passing through pseudomonotone mappings, the boundednessassumption can be omitted; cf. Theorem 2.18 below.


Therefore, by (norm×weak)-continuity of the duality pairing, passing to the limitwith k → ∞ yields

0 ≤ limk→∞

⟨fk −A(v), uk − v

⟩=⟨f −A(v), u − v

⟩. (2.32)

Then we apply again the Minty-trick Lemma 2.13, which gives A(u) = f . Thuseven the whole sequence {uk}k∈N converges weakly.

If A is d-monotone, we can refine (2.31) used for v := u as follows:(d(‖uk‖)− d(‖u‖))(‖uk‖ − ‖u‖) ≤ ⟨A(uk)−A(u), uk − u

⟩=⟨fk −A(u), uk − u

⟩→ ⟨f −A(u), u− u

⟩= 0, (2.33)

which gives ‖uk‖ → ‖u‖ because d : R → R is increasing. Hence uk → u byTheorem 1.2. In other words, A−1 is continuous.

The point (iii): By (2.2) one has for any A(u1) = f1 and A(u2) = f2 theestimate

ζ(‖u1 − u2‖

)‖u1 − u2‖ ≤ ⟨A(u1)−A(u2), u1 − u2

⟩= 〈f1 − f2, u1 − u2〉 ≤ ‖f1 − f2‖∗‖u1 − u2‖ (2.34)

so that ζ(‖u1 − u2‖) ≤ ‖f1 − f2‖∗. By the assumed properties of ζ, the inversemapping A−1 is uniformly continuous. The case of strong monotonicity is obvious.

�

Lemma 2.15. Any monotone mapping A : V → V ∗ is locally bounded in the sense:

∀u∈V ∃ε > 0 ∃M ∈R+ ∀v∈V : ‖v−u‖ ≤ ε ⇒ ‖A(v)‖∗ ≤ M. (2.35)

Proof. Suppose the contrary, i.e. (2.35) does not hold at some u ∈ V . Withoutloss of generality, assume u = 0. This means that there is a sequence {vk}, vk → 0,such that ‖A(vk)‖∗ → ∞. Putting ck := 1 + ‖A(vk)‖∗‖vk‖, we can estimate bymonotonicity of A that

⟨A(vk)ck

, v⟩

≤⟨A(vk), vk

⟩+⟨A(v), v − vk

⟩ck

≤ 1 +∥∥A(v)∥∥∗(‖v‖+ ‖vk‖

)→ 1 + ‖A(v)‖∗‖v‖. (2.36)

Replacing v by −v, we can conclude that lim supk→∞ |〈c−1k A(vk), v〉| < +∞

for any v ∈ V . By Banach-Steinhaus’ Theorem 1.1, c−1k ‖A(vk)‖∗ ≤ M . This

means ‖A(vk)‖∗ ≤ Mck = M(1+‖A(vk)‖∗‖vk‖), and then also ‖A(vk)‖∗ ≤M/(1−M‖vk‖) → M , which contradicts the fact that ‖A(vk)‖∗ → ∞. �

Lemma 2.16. Radially continuous monotone mappings are also demicontinuous.


Proof. Take a sequence {uk}k∈N convergent to some u ∈ V . By Lemma 2.15,{A(uk)}k∈N is bounded in V ∗ and, by Banach Theorem 1.7, we can select a subse-quence {A(ukl

)}l∈N converging weakly to some f ∈ V ∗. Then, by the monotonicityof A, 0 ≤ liml→∞〈A(ukl

)−A(v), ukl−v〉 = 〈f −A(v), u−v〉. As v is arbitrary and

we assume radial continuity of A, the Minty-trick Lemma 2.13 yields f = A(u). Asf is thus determined uniquely, even the whole sequence {A(uk)}k∈N must convergeto it weakly. �Proposition 2.17. Let A = A1 + A2 : V → V ∗ be coercive, and A1 be radiallycontinuous and monotone and A2 be totally continuous. Then A is surjective.

Proof. As in the proof of Brezis’ Theorem 2.6, consider uk ∈ Vk the Galerkinapproximations (2.8), i.e. here⟨

A1(uk) +A2(uk), v⟩= 〈f, v〉 ∀v∈Vk, (2.37)

and the a-priori estimate (2.13), and choose a weakly convergent subsequence{uki}i∈N with a limit u ∈ V . Use monotonicity of A1 to write

0 ≤ ⟨A1(vl)−A1(uki), vl−uki

⟩=⟨A1(vl), vl−uki

⟩+⟨A2(uki)−f, vl−uki

⟩(2.38)

for any vl ∈ Vl with l ≤ k. Passing to the limit with i → ∞, it gives

0 ≤ ⟨A1(vl), vl − u⟩+⟨A2(u)− f, vl − u

⟩. (2.39)

Then, by density of⋃

k∈N Vk in V , consider vl → v for v∈V arbitrary, use demi-continuity of A1 (cf. Lemma 2.16), and pass to the limit with l → ∞ to get:

0 ≤ ⟨A1(v), v − u⟩+⟨A2(u)− f, v − u

⟩. (2.40)

Finally, replace v by u + εw with w ∈ V arbitrary and use Minty’s trick as in(2.26)–(2.27) to show that A1(u) +A2(u) = f . �

In principle, if A1 is also bounded, one could use Lemma 2.9 and Corol-lary 2.12 to see that A from Proposition 2.17 is surjective; realize that A2, beingtotally continuous, is certainly bounded. The above direct proof allowed us to avoidthe boundedness assumption of A1. In particular, for A2 = 0, one thus obtains thecelebrated assertion:

Theorem 2.18 (Browder [73] and Minty [286]). Any monotone, radially con-tinuous, and coercive A : V → V ∗ is surjective.

As a very special case, one gets another celebrated result:

Theorem 2.19 (Lax and Milgram [253]6). Let V be a Hilbert space, A : V → V ∗

be a linear continuous operator which is positive definite in the sense 〈Av, v〉 ≥ε‖v‖2 for some ε > 0. Then A has a bounded inverse.

6A usual formulation uses a bounded, positive definite, bilinear form a : V × V → R. Thisform then determines A : V → V ∗ through the identity 〈Au, v〉 = a(u, v).


Sometimes, the following modification of Proposition 2.17 can be advanta-geously applied, obtaining also the strong convergence of Galerkin’s approximatesolutions.

Proposition 2.20. Let A = A1 + A2 : V → V ∗ be coercive, and A1 be monotoneradially continuous and satisfy (2.23), and A2 be demicontinuous and compact.7

Then A is surjective.

Proof. We have the Galerkin identity (2.37) and a subsequence uk ⇀ u, and write⟨A1(uk)−A1(u), uk−u

⟩=⟨A1(uk)−A1(u), vk−u

⟩+⟨A1(uk)−A1(u), uk−vk

⟩=⟨A1(uk)−A1(u), vk−u

⟩+⟨f−A2(uk)−A1(u), uk−vk

⟩=: I

(1)k + I

(2)k . (2.41)

As A1 is monotone, for any ε > 0, then∥∥A1(uk)∥∥∗ =

1

εsup‖v‖≤ε

⟨A1(uk), v

⟩ ≤ 1

εsup‖v‖≤ε

(⟨A1(uk), v

⟩+⟨A1(uk)−A1(v), uk−v

⟩)=

1

εsup‖v‖≤ε

(⟨A1(uk), uk

⟩+⟨A1(v), v−uk

⟩). (2.42)

Now we use that {〈A1(uk), uk〉}k∈N is bounded because 〈A1(uk), uk〉=〈f−A2(uk),uk〉 and the compact mapping A2 is certainly bounded, and also {〈A1(v), v −uk〉; ‖v‖ ≤ ε} is bounded if ε > 0 is small enough because A1 is locally boundedaround the origin due to Lemma 2.15. Thus (2.42) shows that ‖A1(uk)−A1(u)‖∗is bounded, and, choosing vk → u in V , we obtain limk→∞ I

(1)k = 0 in (2.41).

Taking a subsequence such that also A2(uk) converges to some χ ∈ V ∗ (as

we can because A2 is compact), we get I(2)k = 〈f − A2(uk) − A1(u), uk − vk〉 →

〈f − χ − A1(u), u − u〉 = 0. As this limit is determined uniquely, even the whole

sequence {I(2)k }k∈N converges to 0.Using (2.41), by (2.23) we get uk→u. By Lemma 2.16, A1(uk)⇀A1(u). By

demicontinuity of A2, also A2(uk)⇀A2(u). It allows us to pass to the limit in(2.37), obtaining 〈A1(u)+A2(u)−f, v〉=0 for any v∈⋃k∈N Vk, hence A(u)=f . �Remark 2.21 (d-monotone A on a uniformly convex V ). Any d-monotone A : V →V ∗ satisfies (2.23) if V is uniformly convex. Indeed, the premise of (2.23) with〈A(uk)−A(u), uk − u〉 ≥ (d(‖uk‖)− d(‖u‖))(‖uk‖− ‖u‖), cf. (2.1), yields ‖uk‖ →‖u‖. Then, by uniform convexity of V and Theorem 1.2, we get immediatelyuk → u.

The nonconstructivity of Brezis’ Theorem 2.6 pointed out in Remark 2.7 canbe avoided in special situations by using Banach’s fixed-point Theorem 1.12 forthe iterative process

uk = Tε(uk−1) := uk−1 − εJ−1(A(uk−1)− f

), k∈N, u0∈V, (2.43)

7In fact, any demicontinuous and compact A2 is automatically continuous.


if V is a Hilbert space and the linear operator J : V → V ∗ is defined by〈Ju, v〉 := (u, v) with (·, ·) denoting here the inner product in V , cf. Remark 3.10.For weakening of the assumptions by further (constructive) approximation seeExample 2.95.

Proposition 2.22 (Banach fixed-point technique). Let V be a Hilbert space,A : V → V ∗ be strongly monotone, i.e. ζ(r) = δr from (2.2) with δ > 0, and alsoA be Lipschitz continuous, i.e. ‖A(u) − A(v)‖∗ ≤ ‖u − v‖. Then the nonlinearmapping Tε defined by (2.43) is contractive for any ε > 0 satisfying

ε < 2δ/ 2 (2.44)

and the fixed point of Tε, i.e. Tε(u) = u, does exist and obviously solves A(u) = f .

Proof. It holds that8 〈f, J−1f〉 = ‖f‖2∗, so that one has

‖Tε(u)− Tε(v)‖2 =⟨J(u − v)− ε(A(u)−A(v)), u − v − εJ−1(A(u)−A(v))

⟩= ‖u− v‖2 − 2ε

(u−v, J−1(A(u)−A(v))

)+ ε2‖J−1A(u)− J−1A(v)‖2

= ‖u− v‖2 − 2ε〈A(u)−A(v), u − v〉+ ε2‖A(u)−A(v)‖2∗≤ ‖u− v‖2 − 2εδ‖u− v‖2 + ε2 2‖u− v‖2.

The condition (2.44) just guarantees the Lipschitz constant√1− 2εδ + ε2 2 of Tε

to be less than 1. �

2.4 Quasilinear elliptic equations

We will illustrate the above abstract theory on boundary-value problems for thequasilinear 2nd-order partial differential equation

−div(a(x, u,∇u)

)+ c(x, u,∇u) = g (2.45)

considered on a bounded connected Lipschitz domain Ω ⊂ Rn. Here a : Ω × R ×Rn → Rn and c : Ω×R×Rn → R; for more qualification see (2.54) and (2.55a,c)below. Recall that ∇u :=

(∂

∂x1u, . . . , ∂

∂xnu)denotes the gradient of u. More in

detail, (2.45) means

−n∑

i=1

∂

∂xiai(x, u(x),∇u(x)

)+ c(x, u(x),∇u(x)

)= g(x) (2.46)

for x∈Ω but we will rather use the abbreviated form (2.45) in what follows. Forsome systems of the 2nd-order equations see Sect. 6.1 below while higher-orderequations will be briefly mentioned in Sect. 2.4.4. Besides, we will confine ourselvesto data with polynomial-growth; p ∈ (1,+∞) will denote the growth of the leading

8Realize that, for v = J−1f , one has 〈f, J−1f〉 = 〈f, v〉 = 〈Jv, v〉 = (v, v) = ‖v‖2 = ‖f‖2∗.

2.4. Quasilinear elliptic equations 43

nonlinearity a(x, u, ·) which essentially determines the setting and the other dataqualification. Also, a(x, u, ·) will be assumed to behave monotonically, cf. (2.65),which is related to the adjective elliptic. For the linear case a(x, r, s) = As, themonotonicity (2.65) and coercivity (2.92a) below implies the matrix A is positivedefinite, which is what is conventionally called “elliptic”, contrary to A indefinite(resp. semidefinite) which is addressed as hyperbolic (resp. parabolic).

Convention 2.23 (Omitting x-variable). For brevity, we will often write a(x, u,∇u)instead of a(x, u(x),∇u(x)) (as we already did in (2.45)) or sometimes evena(u,∇u) if the dependence on x is automatic; hence, in fact, Na(u,∇u) =a(u,∇u). Thus, e.g.

∫Ωc(u,∇u)v dx will mean

∫Ωc(x, u(x),∇u(x))v(x) dx.

2.4.1 Boundary-value problems for 2nd-order equations

The equation (2.45) may admit very many solutions, which indicates some missingrequirements. This is usually overcome by a boundary condition to be prescribedfor the solution on the boundary Γ:= ∂Ω of the domain Ω.

One option is to prescribe simply the trace u|Γ of u on the boundary, i.e.

u|Γ = uD on Γ (2.47)

with uD a fixed function on Γ. This condition is referred to as a Dirichlet boundarycondition.

Having in mind the equation (2.45), the alternative natural possibility is toprescribe a local equation for the “boundary flux” ν · a, i.e.

ν · a(x, u,∇u) + b(x, u) = h on Γ (2.48)

where ν = (ν1, . . . , νn) denotes the unit outward normal to Γ and h : Γ → R andb : Γ × R → R are given functions qualified later. More in detail, (2.48) means∑n

i=1 νi(x) ai(x, u(x),∇u(x)) + b(x, u(x)) = h(x) for x ∈ Γ. This condition isreferred to as a (nonlinear) Newton boundary condition or sometimes also a Robincondition. If b = 0, it is called a Neumann boundary condition.

One can still think about a combination of (2.47) and (2.48) on various partsof Γ. For this, let us divide (up to a zero-measure set) the boundary Γ on twodisjoint open parts ΓD and ΓN such that measn−1

(Γ \ (ΓD ∪ ΓN)

)= 0, and then

consider so-called mixed boundary conditions

u|Γ = uD on ΓD, (2.49a)

ν · a(x, u,∇u) + b(x, u) = h on ΓN. (2.49b)

As either ΓD or ΓN may be empty, (2.49) covers also (2.48) and (2.47), respectively.Completing the equation (2.45) with the boundary conditions (2.47) (resp.

(2.48) or (2.49)), we will speak about a Dirichlet (resp. Newton or mixed)boundary-value problem. One can have an idea to seek a so-called classical so-lution u of it, i.e. such u ∈ C2(Ω) satisfying the involved equalities everywhere on


Ω and Γ. This requires, however, very strong data qualifications both for a, b, andc and for Ω itself. Therefore, modern theories rely on a natural generalization ofthe notion of the solution. In this context, ultimate requirements on every sensibledefinition are9:

1. Consistency: Any classical solution to the boundary-value problem in questionis the generalized solution.

2. Selectivity: If all data are smooth and if the generalized solution belongs toC2(Ω), then it is the classical solution. Moreover, speaking a bit vaguely, inqualified cases the generalized solution is unique.

2.4.2 Weak formulation

Here, the generalized solution will arise from a so-called weak formulation of theboundary-value problem, which is the most frequently used concept and whichjust fits to the pseudomonotonicity approach. Later, we will present some otherconcepts, too.

For the full generality, we will treat the mixed boundary conditions (2.49).The weak formulation of (2.45) with (2.49) arises as follows:

Step 1: Multiply the differential equation, i.e. here (2.45), by a test function v.

Step 2: Integrate it over Ω.

Step 3: Use Green’s formula (1.54), here with z = a(x, u,∇u).

Step 4: Substitute the Newton boundary condition, i.e. here (2.49b), into theboundary integral, i.e. here

∫ΓN

v(z · ν) dS =∫ΓN(ν · a(x, u,∇u))v dS in

(1.54), while by considering v|ΓD = 0, the integral over ΓD simply van-ishes.

This procedure looks here as∫Ω

(− div

(a(x, u,∇u)

)+ c(x, u,∇u)

)v dx

Green’sformula

=

∫Ω

a(x, u,∇u) · ∇v+c(x, u,∇u)v dx−∫Γ

(ν ·a(x, u,∇u)

)v dS

boundaryconditions

=

∫Ω

a(x, u,∇u) · ∇v+c(x, u,∇u)v dx+

∫Γ

(b(x, u)− h(x)

)v dS. (2.50)

Realizing still that the left-hand side in (2.50) is just∫Ωgv dx, we come to the

integral identity∫Ω

a(x, u,∇u) · ∇v+c(x, u,∇u)v dx+

∫ΓN

b(x, u)v dS=

∫Ω

gv dx+

∫ΓN

hv dS. (2.51)

9See [360, Remark 5.3.8] or [370] for some examples of unsuitable concepts of so-called“measure-valued” solutions, cf. also DiPerna [124] or Illner and Wick [210].


As declared, we confine ourselves to a p-polynomial growth, cf. (2.55a) below, andthen it is natural to seek the weak solution in the Sobolev space W 1,p(Ω). It leadsto the following definition:

Definition 2.24. We call u ∈ W 1,p(Ω) a weak solution to the mixed boundary-valueproblem (2.45) and (2.49) if u|ΓD = uD and if the integral identity (2.51) holds forany v ∈ W 1,p(Ω) with v|ΓD = 0.

The above 4-step procedure to derive (2.51) guarantees automatically itsconsistency. On the other hand, its selectivity is related to the important fact thatthe space V of test-functions v’s, i.e.

V ={v∈W 1,p(Ω); v|ΓD = 0

}, (2.52)

is sufficiently rich, the restriction of v on ΓD being compensated by direct involve-ment of the boundary condition (2.49a) in Definition 2.24:

Proposition 2.25 (Selectivity of the weak-solution definition). Let a ∈C1(Ω × R × Rn;Rn), c ∈ C0(Ω × R × Rn), and b ∈ C0(ΓN × R), g ∈ C(Ω), andh ∈ C(ΓN). Then any weak solution u ∈ C2(Ω) is the classical solution.

Proof. Put v ∈ V into (2.51) and use Green’s formula (1.54). One gets∫Ω

(div a(x, u,∇u)− c(x, u,∇u) + g

)v dx

+

∫ΓN

(h− b(x, u)− ν·a(x, u,∇u)

)v dS = 0. (2.53)

Considering v|Γ = 0, the boundary integral in (2.53) vanishes. As v is otherwisearbitrary, one deduces that (2.45) holds a.e., and hence even everywhere in Ω dueto the assumed smoothness of a and c.10 Hence, the first integral in (2.53) vanishes.Then, putting a general v∈V into (2.53) shows the latter boundary condition in(2.49) valid,11 while the former one is directly involved in Definition 2.24. �

The important issue now is to set up basic data qualification to give a senseto all integrals in (2.51). Recall that we keep the permanent assumption Ω to bea bounded Lipschitz domain (so that, in particular, ν is defined a.e. on Γ) and ΓD

and ΓN are open in Γ (hence, in particular, measurable). To ensure measurabilityof integrands on the left-hand side of (2.51) we must assume:

ai, c : Ω× R× Rn → R , b : Γ× R → R are Caratheodory functions, (2.54)

for i = 1, . . . , n; this means measurability in x and continuity in the other variables.The further ultimate requirement is integrability of all integrands on the left-hand

10Here we use the fact that the set of test functions is sufficiently rich, namely that W 1,p0 (Ω)

is dense in L1(Ω); cf. Theorem 1.25 and the well-known fact that C∞0 (Ω) is dense in L1(Ω).

11Here the important fact is that the set {v|ΓN ; v ∈ V } is dense in L1(ΓN). This is guaranteedby the assumption that ΓN is open in Γ.


side of (2.51). This, and some continuity requirements needed further, lead us toassume the growth conditions on the nonlinearities a, b, and c:

|a(x, r, s)| ≤ γ(x) + C|r|(p∗−ε)/p′+ C|s|p−1 for some γ∈Lp′

(Ω) , (2.55a)

| b(x, r) | ≤ γ(x) + C|r|p#−ε−1 for some γ∈Lp#′(Γ), (2.55b)

|c(x, r, s)| ≤ γ(x) + C|r|p∗−ε−1 + C|s|p/p∗′for some γ∈Lp∗′

(Ω). (2.55c)

Let us recall the notation of the prime denoting the conjugate exponents (i.e.,e.g., p′ = p/(p−1), cf. (1.20)) and the continuous (resp. compact) embeddingW 1,p(Ω)⊂Lp∗

(Ω) (resp. W 1,p(Ω)�Lp∗−ε(Ω) with ε>0), cf. Theorem 1.20. More-

over, the trace operator u �→ u|Γ maps W 1,p(Ω) into Lp#

(Γ) continuously and into

Lp#−ε(Γ) compactly, cf. Theorem 1.23. For p∗ and p# see (1.34) and (1.37).

Convention 2.26. For p > n, the terms |r|+∞ occurring in (2.55) are to be under-stood such that |a(x, ·, s)|, |b(x, ·)|, and |c(x, ·, s)| may have arbitrary fast growthif |r| → ∞.

In view of Theorem 1.27, the growth conditions (2.55) are designed so thatrespectively

Na :W1,p(Ω)×Lp(Ω;Rn) → Lp′

(Ω;Rn) is (weak××norm,norm)-continuous, (2.56a)

u �→ Nb(u|Γ) :W 1,p(Ω) → Lp#′(Γ) is (weak,norm)-continuous, (2.56b)

Nc :W1,p(Ω)×Lp(Ω;Rn) → Lp∗′

(Ω) is (weak×norm,norm)-continuous. (2.56c)

In particular, for u, v ∈ W 1,p(Ω), the integrands a(x, u,∇u) · ∇v and c(x, u,∇u)voccurring in (2.51) belong to L1(Ω) while b(x, u|Γ)v|Γ belongs to L1(Γ).

Furthermore, we will also suppose the right-hand side qualification:

g ∈ Lp∗′(Ω), h ∈ Lp#′

(Γ). (2.57)

Note that (2.57) ensures gv ∈ L1(Ω) and hv|Γ ∈ L1(Γ) for v ∈ W 1,p(Ω), hence(2.51) has a good sense. Moreover, we must qualify uD occurring in the Dirichletboundary condition (2.49a). The simplest way is to assume

∃w ∈ W 1,p(Ω) : uD = w|Γ. (2.58)

Then, considering V from (2.52) equipped by the norm (1.30b) denoted simply by‖ · ‖, we define A : W 1,p(Ω) → V ∗ and f ∈ V ∗ simply by⟨

A(u), v⟩:= left-hand side of (2.51), (2.59)⟨

f, v⟩

:= right-hand side of (2.51). (2.60)

Moreover, referring to (2.58), let us define A0 : V → V ∗ by

A0(u) = A(u + w). (2.61)


Note that A0 has again the form of A from (2.51) but the nonlinearities a, b, and care respectively replaced by a0, b0, and c0 given by a0(x, r, s) := a(x, r+w(x), s+∇w(x)), b0(x, r) := b(x, r + w(x)), and c0(x, r, s) := c(x, r + w(x), s + ∇w(x)),and these nonlinearities satisfy (2.54)–(2.55) if w ∈ W 1,p(Ω) and if the originalnonlinearities a, b, and c satisfy (2.54)–(2.55). Note also that for zero (or none)Dirichlet boundary conditions, one can assume w = 0 in (2.58) and then A0 ≡ A|V(or simply A0 ≡ A).

Note that, indeed, f ∈ V ∗ because of the obvious estimate

‖f‖∗ = sup‖v‖≤1

(∫Ω

gv dx+

∫ΓN

hv dS)≤ sup‖v‖≤1

(‖g‖Lp∗′(Ω)‖v‖Lp∗(Ω)

+ ‖h‖Lp#

′(ΓN)

‖v‖Lp#(ΓN

)≤ N1‖g‖Lp∗′(Ω) +N2‖h‖Lp#

′(ΓN)

(2.62)

where N1 is the norm of the embedding operator W 1,p(Ω) → Lp∗(Ω) and N2 is the

norm of the trace operator v �→ v|ΓN : W 1,p(Ω) → Lp#

(ΓN). By similar arguments,(2.54) and (2.55) ensures A(u) ∈ V ∗, cf. Lemma 2.31 below.

Proposition 2.27 (Shift for non-zero Dirichlet condition). The abstractequation (2.6) for A0 has a solution u0 ∈ V , i.e. A0(u0) = f , if and only ifu = u0 + w ∈ W 1,p(Ω) is the weak solution to the boundary-value problem (2.45)and (2.49) in accord to Definition 2.24.

Proof. We obviously have f = A0(u0) = A0(u−w) = A(u−w+w) = A(u), hencethe assertion immediately follows by the definition (2.59)–(2.60). �

Remark 2.28 (Why both u and v are from V ). In principle, Definition 2.24 couldwork with v ∈ Z := W 1,∞(Ω), or even with v’s smoother; the selectivity Propo-sition 2.25 would hold as far as density of Z in V would be preserved, as used inSection 2.5 below. The choice of v’s from the same space where the solution u issupposed to live, i.e. here V , is related to the setting A : V → Z∗ which is fittedwith the pseudomonotone-mapping concept only for Z = V .

Remark 2.29 (Why both ΓD and ΓN are assumed open). In principle, Definition 2.24as well as the existence Theorem 2.36 below could work for ΓD and ΓN only mea-surable. However, we would lose the connection to the original problem, cf. Propo-sition 2.25: indeed, one can imagine ΓD measurable dense in Γ and ΓN of a positivemeasure. Then, for p > n, v|Γ ∈ C(Γ) and the condition v|ΓD = 0 would im-ply v|Γ = 0, so that the ΓN-integrals in (2.51) vanish and the Newton boundarycondition on ΓN in (2.49b) would be completely eliminated.

Remark 2.30 (Integral balance). The equation (2.45) is a differential alternativeto the integral balance∫

O

c(x, u,∇u)− g(x) dx =

∫∂O

a(x, u,∇u) · ν dS (2.63)


for any test volumeO ⊂ Ω with O ⊂ Ω and a smooth boundary ∂O with the normalν = ν(x). Obviously, one is to identify c as the balanced quantity (depending onu and ∇u) while a as a flux of this quantity12, and then (2.63) just says that theoverall production of this quantity over the arbitrary test volume O is balancedby the overall flux through the boundary ∂O, cf. Figure 4. The philosophy thatintegral form (2.63) of physical laws is more natural than their differential form(2.45) was pronounced already by David Hilbert13. The weak formulation (2.51)implicitly includes, besides information about the boundary conditions, also (2.63).Indeed, it suffices to take v in (2.51) as some approximation of the characteristicfunction χO (which itself does not belong to W 1,p(Ω), however), e.g. vε withvε(x) := (1 − dist(x,O)/ε)+, and then to pass ε ↘ 0. This limit passage is,however, legal only if x �→ a(x, u,∇u) is sufficiently regular near ∂O or, in ageneral case, it holds only in some “generic” sense; cf. e.g. Exercise 2.63.

Ω

O

dx

dS

a·ν−a·ν

a = a(u,∇u)

ac− g =c(u,∇u)−g

Figure 4. Illustration to balancing the normal flux a·ν through the bound-ary of a test volume O and the production c inside this volume.

2.4.3 Pseudomonotonicity, coercivity, existence of solutions

In view of Theorem 2.6 with Proposition 2.27, we are to show pseudomonotonicityof A0 : V → V ∗. For simplicity, we can prove it for A as W 1,p(Ω) → W 1,p(Ω)∗,which, by Lemma 2.11(ii), implies pseudomonotonicity of A0 : W 1,p(Ω) →W 1,p(Ω)∗, and then obviously also of A0 : V → V ∗. Let us prove (2.3a) and(2.3b) respectively in the following lemmas.

Lemma 2.31 (Boundedness of A). The assumptions (2.54) and (2.55) ensure(2.3a), i.e. A : W 1,p(Ω) → W 1,p(Ω)∗ bounded.

Proof. We prove A({u ∈ W 1,p(Ω); ‖u‖ ≤ ρ}) bounded in W 1,p(Ω)∗ for any ρ > 0.

Here, ‖ · ‖ and ‖ · ‖∗ will denote the norms in W 1,p(Ω) and W 1,p(Ω)∗, respectively.Indeed, we can estimate

12In concrete situations, the dependence of a on ∇u may result from a (nonlinear) Fick’s,Fourier’s, or Darcy’s law.

13Explicitly, it can be found in his famous Mathematical problems [202, 19th problem]: “Hasnot every . . . variational problem a solution, provided . . . if need be that the notion of a solutionshall be suitably extended?”


sup‖u‖≤ρ

‖A(u)‖∗ = sup‖u‖≤ρ

sup‖v‖≤1

〈A(u), v〉

= sup‖u‖≤ρ

sup‖v‖≤1

∫Ω

a(u,∇u) · ∇v + c(u,∇u)v dx+

∫ΓN

b(u)v dS

≤ sup‖u‖≤ρ

sup‖v‖≤1

∥∥a(u,∇u)∥∥Lp′(Ω;Rn)

∥∥∇v∥∥Lp(Ω;Rn)

+∥∥c(u,∇u)

∥∥Lp∗′(Ω)

∥∥v∥∥Lp∗(Ω)

+∥∥b(u)∥∥

Lp#′(ΓN)

∥∥v∥∥Lp#(ΓN)

≤ sup‖u‖≤ρ

∥∥a(u,∇u)∥∥Lp′(Ω;Rn)

+N1

∥∥c(u,∇u)∥∥Lp∗′(Ω)

+N2

∥∥b(u)∥∥Lp#

′(ΓN)

(2.64)

where N1 and N2 are as in (2.62). In view of (2.55), it is bounded uniformly for uranging over a bounded set in W 1,p(Ω). �

Further, we still have to strengthen our data qualification. The crucial as-sumption we must make for pseudomonotonicity of A is the so-called monotonicityin the main part:

∀(a.a.) x∈Ω ∀r∈R ∀s, s∈Rn :(a(x, r, s)− a(x, r, s)

) · (s− s) ≥ 0. (2.65)

To cover as many situations as possible, we distinguish three cases in accordancewith whether c(x, r, ·) is constant, linear, or nonlinear, respectively.Lemma 2.32 (The implication (2.3b)). Let the assumptions (2.54) and (2.55)be valid, let a satisfy (2.65), and let one of the following three cases hold: c isindependent of s, i.e. for some c : Ω× R → R,

c(x, r, s) = c(x, r) , (2.66)

or c is linearly dependent on s, i.e. for some c : Ω× R → Rn,

c(x, r, s) = c(x, r) · s, (2.67)

or c is generally dependent on s but the strict monotonicity “in the main part”and coercivity of a(x, r, ·) hold and the growth of c(x, ·, ·) is further restricted:

(a(x, r, s)− a(x, r, s)) · (s− s) = 0 =⇒ s = s, (2.68a)

∀s0∈Rn : lim|s|→∞

a(x, r, s)·(s−s0)

|s| = +∞ uniformly for r bounded, (2.68b)

∃γ∈Lp∗′+ε(Ω)∃C∈R : |c(x, r, s)| ≤ γ(x)+C|r|p∗−ε−1+C|s|(p−ε)/p∗′(2.68c)

with Convention 2.26 in mind. Then A : W 1,p(Ω) → W 1,p(Ω)∗ satisfies (2.3b).

Remark 2.33. Obviously, (2.66) together with the growth condition (2.55c) imply|c(x, r)| ≤ γ(x) +C|r|p∗−ε−1 with γ as in (2.55c). A bit more difficult is to realize


that (2.67) together with the growth condition (2.55c) imply that c : Ω×R → Rn

has to satisfy

|c(x, r)| ≤ γ(x) + C|r|p∗/q−ε1 with γ∈Lq+ε1(Ω) and some ε1 > 0,

where q =

{ np

np− 2n+ pif p < n,

p′ if p ≥ n.(2.69)

This condition together with the structural condition (2.67) now guarantees

Nc :W1,p(Ω)×Lp(Ω;Rn)→L(p∗−ε)′(Ω) is (weak×weak,weak)-continuous. (2.70)

Eventually, note that the growth condition (2.68c) strengthens (2.55c) and is de-signed so that, for some ε > 0 (depending on ε used in (2.68c)),

Nc : W1,p(Ω)×Lp(Ω;Rn) → Lp∗′+ε(Ω) is (weak×norm,norm)-continuous. (2.71)

Proof of Lemma 2.32. Let us take uk ⇀ u in W 1,p(Ω) and assume that

lim supk→∞

〈A(uk), uk − u〉 ≤ 0. (2.72)

We are to show that lim infk→∞〈A(uk), uk−v〉 ≥ 〈A(u), u−v〉 for any v∈W 1,p(Ω).To distinguish between the highest and the lower-order terms, we define B(w, u)∈W 1,p(Ω)∗ by⟨

B(w, u), v⟩:=

∫Ω

a(x,w,∇u) · ∇v + c(x,w,∇w)v dx+

∫ΓN

b(x,w)v dS (2.73)

for u,w ∈ W 1,p(Ω); recall the Convention 2.23. Obviously, A(u) = B(u, u).Let us put uε = (1−ε)u + εv, ε ∈ [0, 1]. Monotonicity (2.65) implies

〈B(uk, uk)−B(uk, uε), uk − uε〉 ≥ 0. Then, just by simple algebra,

ε⟨A(uk), u− v

⟩ ≥ − ⟨A(uk), uk − u⟩

+⟨B(uk, uε), uk − u

⟩+ ε⟨B(uk, uε), u− v

⟩. (2.74)

Let us assume, for a moment, that we have proved

limk→∞

⟨B(uk, v), uk − u

⟩= 0, (2.75)

w-limk→∞

B(uk, v) = B(u, v) (the weak limit in W 1,p(Ω)∗), (2.76)

and use them here for v = uε to pass successively to the limit in the right-hand-sideterms of (2.74). Using also (2.72), we thus obtain

ε lim infk→∞

⟨A(uk), u−v

⟩ ≥ − lim supk→∞

⟨A(uk), uk−u

⟩+ lim

k→∞⟨B(uk, uε), uk−u

⟩+ ε lim

k→∞⟨B(uk, uε), u−v

⟩ ≥ ε⟨B(u, uε), u−v

⟩.


Divide it by ε > 0. Then the limit passage ε → 0 gives uε → u strongly so that weget B(u, uε) → B(u, u) even strongly14, which results in lim infk→∞〈A(uk), u −v〉 ≥ 〈B(u, u), u− v〉 = 〈A(u), u− v〉.

Then, by using the monotonicity in the main part (2.65) once again, now as〈B(uk, uk) − B(uk, u), uk − u〉 ≥ 0, and by using also (2.75) now with v = u, wecan claim that

lim infk→∞

⟨A(uk), uk−v

⟩ ≥ lim infk→∞

⟨A(uk), uk−u

⟩+ lim inf

k→∞⟨A(uk), u−v

⟩= lim

k→∞⟨B(uk, u), uk−u

⟩+ lim inf

k→∞⟨B(uk, uk)−B(uk, u), uk−u

⟩+ lim inf

k→∞⟨A(uk), u − v

⟩ ≥ 〈A(u), u− v〉, (2.77)

which is just the conclusion of (2.3b).Thus it remains to prove (2.75) and (2.76). Since uk ⇀ u in W 1,p(Ω) �

Lp∗−ε(Ω), we have uk → u in Lp∗−ε(Ω). Similarly, uk|Γ → u|Γ in Lp#−ε(Γ). Then,by the continuity of the Nemytskiı mappings induced by a(·,∇v) and b, we get

a(uk,∇v) → a(u,∇v) in Lp′(Ω;Rn), cf. (2.56a), and b(uk) → b(u) in Lp#′

(Γ);cf. (2.56b) together with (1.36b); recall again Convention 2.23. Hence, realizing

that ∇(uk−u) ⇀ 0 in Lp(Ω;Rn) and (uk−u)|Γ ⇀ 0 in Lp#

(ΓN), one gets∫Ω

a(uk,∇v) · ∇(uk − u) dx+

∫ΓN

b(uk)(uk − u) dS → 0. (2.78)

By the same reasons, for any z ∈ W 1,p(Ω), we have also∫Ω

a(uk,∇v)·∇z dx+

∫ΓN

b(uk)z dS →∫Ω

a(u,∇v)z dx+

∫ΓN

b(u)z dS. (2.79)

As to the term c, we will distinguish the above suggested three cases.

The case (2.66): By the continuity of the Nemytskiı mappings induced by c, onehas c(uk) → c(u) in Lp∗′

(Ω). Therefore, realizing that uk − u ⇀ 0 in Lp∗(Ω), one

gets∫Ω c(uk)(uk − u) dx → 0. Adding it with (2.78), one gets

〈B(uk, v), uk − u〉 :=

∫Ω

(a(uk,∇v) · ∇(uk − u)

+ c(uk)(uk − u))dx+

∫ΓN

b(uk)(uk − u) dS → 0, (2.80)

which proves (2.75). Similarly,∫Ω c(uk)zdx → ∫

Ω c(uk)zdx, which, together with(2.79), gives just (2.76).

The case (2.67): Here we have a certain reserve in the growth, cf. (2.70), and canthus exploit the compactness of the embeddingW 1,p(Ω) � Lp∗−ε(Ω) to use uk → u

14Here we use the continuity of the Nemytskiı mapping Na◦u with a◦u : (x, s) �→ a(x, u(x), s).


strongly in Lp∗−ε(Ω). Also, we can use c(uk) → c(u) in Lq+ε1(Ω) with q from (2.69)and some ε1 > 0 (depending on ε); note that (q+ ε1)

−1 + p−1+(p∗− ε)−1 ≤ 1 if εis small enough depending on the chosen ε1. As ∇uk ⇀ ∇u in Lp(Ω;Rn), we canpass to the limit in the c-term:∫

Ω

c(uk) · ∇uk(uk − u) dx → 0. (2.81)

Adding it with (2.78), one gets (2.75). Similarly,∫Ω c(uk) · ∇ukz dx → ∫

Ω c(uk) ·∇uz dx, which, together with (2.79), gives just (2.76).

The case (2.68): We already showed that uk → u in Lp∗−ε(Ω). In view of theboundedness (2.71) of {c(uk,∇uk)}k∈N in Lp∗′+ε(Ω), we obviously have∫

Ω

c(uk,∇uk)(uk − u) dx → 0 . (2.82)

Adding it with (2.78), one gets (2.75).To prove (2.76), we need to show a convergence of ∇uk to ∇u in a better

mode than the weak one only. Let us denote

ak(x) :=(a(x, uk(x),∇uk(x)) − a(x, uk(x),∇u(x))

) · ∇(uk(x)− u(x)). (2.83)

By the monotonicity (2.65), it holds

0 ≤ lim supk→∞

∫Ω

ak(x) dx = lim supk→∞

⟨B(uk, uk)−B(uk, u), uk − u

⟩= lim sup

k→∞

⟨A(uk), uk − u

⟩− limk→∞

⟨B(uk, u), uk − u

⟩ ≤ 0; (2.84)

note that the last limit superior is non-positive by assumption while the last limitequals zero by (2.75) with v := u. This implies that ak → 0 in the measure so thatwe can select a subsequence such that

ak(x) → 0 (2.85)

for a.a. x ∈ Ω. As uk → u strongly in Lp∗−ε(Ω), by Proposition 1.13(ii)–(iii) wecan further select a subsequence that also

uk(x) → u(x) (2.86)

for a.a. x ∈ Ω. Take x ∈ Ω such that both (2.85) and (2.86) hold and also ∇u(x),∇uk(x), k ∈ N, and γ(x) from (2.55a) are finite, and a(x, ·, ·) is continuous. Ifthe sequence {∇uk(x)}k∈N would be unbounded, then the coercivity (2.68b) usedfor s0 = ∇u(x) would yield lim supk→∞(a(x, uk(x),∇uk(x)) − a(x, uk(x), s0)) ·(∇uk(x) − s0) = +∞, which would contradict (2.85). Therefore, we can take asuitable s ∈ Rn and a (for a moment sub-) sequence such that ∇uk(x) → s in Rn.


By (2.85) and (2.86) and the continuity of a(x, ·, ·), cf. (2.54), we can pass tothe limit in (2.83), which yields(

a(x, u(x), s) − a(x, u(x),∇u(x))) · (s−∇u(x)

)= 0. (2.87)

By the strict monotonicity (2.68a), we get s = ∇u(x). As s is determined uniquely,even the whole sequence {∇uk(x)}k∈N converges to s.15 Then

c(uk,∇uk) → c(u,∇u) a.e. in Ω. (2.88)

By Holder’s inequality, for any measurable S ⊂ Ω, we can estimate∫S

∣∣c(uk,∇uk)−c(u,∇u)∣∣p∗′dx ≤ ∥∥c(uk,∇uk)−c(u,∇u)

∥∥Lp∗′+ε(Ω)

measd(S)1+p∗′/ε.

(2.89)

Further, we realize that the sequence {c(uk,∇uk)−c(u,∇u)}k∈N is bounded inLp∗′+ε(Ω) thanks to the assumption (2.68c). Thus (2.89) verifies the equi-absolute-continuity of the collection {|c(uk,∇uk)−c(u,∇u)|p∗′}k∈N, cf. (1.28). By Dunford-Pettis’ theorem 1.16(ii), {|c(uk,∇uk)−c(u,∇u)|p∗′}k∈N is also uniformly inte-grable and, since it converges to 0 a.e. due to (2.88), by Vitali’s theorem 1.17,|c(uk,∇uk)−c(u,∇u)|p∗′ → 0 in L1(Ω), i.e.

c(uk,∇uk) → c(u,∇u) in Lp∗′(Ω). (2.90)

As the limit c(u,∇u) is determined uniquely, even the whole sequence (not onlythat one selected for (2.85)–(2.86)) must converge. Then (2.76) follows by joining(2.90) with (2.79). �

Note that, as always p∗′+ε > 1, (2.90) also proves that

c(uk,∇uk) ⇀ c(u,∇u) in Lp∗′+ε(Ω). (2.91)

By the same technique one can also prove a(uk,∇uk) ⇀ a(u,∇u) weakly inLp′

(Ω;Rn). We however did not need this fact in the above proof.

Remark 2.34 (Critical growth in lower-order terms). The above theorem and itsproof permits various modifications: If b(x, ·) is monotone, then the splitting (2.73)can involve b(u) instead of b(w), which allows for borderline growth of b, i.e. (2.55b)with ε = 0. Similarly, if c = c(x, r) as in (2.66) but with c(x, ·) is monotone, then(2.73) can involve c(u) instead of c(w,∇w), and (2.55c) with ε = 0 suffices. Mod-ification of the basic space V in these cases would allow for even a super-criticalgrowth, cf. (2.128). The growth restriction can also be eliminated if a maximumprinciple, guaranting L∞-estimates, is at our disposal, which unfortunately can beexpected only in special cases like the equation Δu = |∇u|2 or (6.73) below.

15The fact that we do not need to select a subsequence at every x in question is importantbecause the set of all such x’s should have the full measure in Ω and thus cannot be countable.


Lemma 2.35 (The coercivity (2.5)). Let the following coercivity hold:

∃ε1, ε2 > 0, k1∈L1(Ω) : a(x, r, s)·s+ c(x, r, s)r ≥ ε1|s|p+ε2|r|q−k1(x), (2.92a)

∃c1 < +∞ ∃k2∈L1(Γ) : b(x, r)r ≥ −c1|r|q1 − k2(x), (2.92b)

for some 1 < q1 < q ≤ p. Then A : W 1,p(Ω) → W 1,p(Ω)∗ is coercive.

Proof. We use the Poincare inequality in the form (1.55), i.e. ‖u‖W 1,p(Ω) ≤C

P(‖∇u‖Lp(Ω;Rn) + ‖u‖Lq(Ω)), which implies

‖u‖qW 1,p(Ω) ≤ 2q−1CqP

(‖∇u‖qLp(Ω;Rn) + ‖u‖qLq(Ω)

)≤ Cp,q

(1 + ‖∇u‖pLp(Ω;Rn) + ‖u‖qLq(Ω)

). (2.93)

Also, by Young’s inequality and boundedness of the trace operator16 u �→ u|Γ :W 1,p(Ω) → Lq(Γ) (let N denote its norm), we use the estimate

∥∥u∥∥q1Lq1(Γ)

=

∫Γ

|u|q1dS ≤∫Γ

ε|u|q+CεdS ≤ εN q∥∥u∥∥q

W 1,p(Ω)+Cεmeasn−1(Γ) (2.94)

with ε > 0 arbitrarily small and Cε < +∞ chosen accordingly; cf. (1.22) withq/q1 > 1 in place of p. Then (2.92) implies the estimate

⟨A(u), u

⟩ ≥∫Ω

(ε1|∇u|p + ε2|u|q − k1

)dx−

∫Γ

(c1|u|q1 + k2

)dS

≥ min(ε1, ε2)

(‖u‖qW 1,p(Ω)

Cp,q− 1

)− ∥∥k1∥∥L1(Ω)

− εN q∥∥u∥∥q

W 1,p(Ω)− Cεmeasn−1(Γ)−

∥∥k2∥∥L1(Γ). (2.95)

When one chooses ε < min(ε1, ε2)/(Cp,qNq) and realizes that q > 1, the coercivity

(2.5) of A, i.e. lim‖u‖W1,p(Ω)→∞〈A(u), u〉 = +∞, is shown. �

Theorem 2.36 (Leray-Lions [257]). Let (2.54), (2.55), (2.57), (2.58), (2.65), and(2.92) be valid and at least one of the conditions (2.66) or (2.67) or (2.68) be valid,then the boundary-value problem (2.45)–(2.49) has a weak solution.

Proof. Lemmas 2.31, 2.32, and 2.35 provedA : W 1,p(Ω) → W 1,p(Ω)∗ pseudomono-tone and coercive. These properties are inherited by A0 : V → V ∗, cf. alsoLemma 2.11(ii). Then we use Theorem 2.6 with Proposition 2.27. �

Remark 2.37 (Coercivity (2.68b)). Note that the coercivity (2.92a) together with(2.55a) and (2.68c) imply the coercivity (2.68b) because

a(x, r, s)·(s− s0) ≥ ε1|s|p + ε2|r|q − k1(x) − c(x, r, s)r − a(x, r, s)·s0 (2.96)

16Note that always q ≤ p < p#.


for such x ∈ Ω that k1(x) is finite. Realizing that s �→ −c(x, r, s)r has a maximaldecay as −|s|(p−ε)/p∗′

due to (2.68c) and s �→ −a(x, r, s) · s0 maximal decay as−|s|p−1 due to (2.55a), the estimate (2.96) shows that s �→ a(x, r, s) · (s− s0) hasthe p-growth uniformly with respect to r bounded because ε > 0 and p∗′ ≥ 1.

Remark 2.38 (Necessity of monotonicity of a(x, r, ·)). Boccardo and Dacorogna[55] showed that monotonicity of a(x, r, ·) is necessary for pseudomonotonicity ofthe mapping A(u) = −div a(x, u,∇u).

Remark 2.39 (Necessity of Leray-Lions’ condition (2.65), (2.68)). If a lower-orderterm c is present, the necessity of strict monotonicity of a(x, r, ·) for the pseu-domonotonicity was shown by Gossez and Mustonen [184].17 It is worth observingthat, for c(x, r, ·) not affine, the mapping u �→ c(u,∇u) : W 1,p(Ω) → W 1,p(Ω)∗,although representing a lower-order term, is neither totally continuous18 norpseudomonotone but it is still compact, cf. Exercise 2.64, and, when added tou �→ −div a(u,∇u), it may result in a pseudomonotone mapping.

Remark 2.40 (General right-hand sides). The functional f : v �→ ∫Ωgv dx +∫

ΓNhv dS we considered, cf. (2.60), is not the general form of a functional

f ∈ W 1,p(Ω)∗. In fact, W 1,p(Ω)∗ would allow g and h to be certain distributionson Ω and Γ, respectively. For example, if p > n, we have a dense and continuousembedding W 1,p(Ω) ⊂ C(Ω) (resp. the trace operator W 1,p(Ω) → C(Γ)), hence-forth the functional f : v �→ ∫

Ωv μ(dx) +

∫Γ v η(dS) with measures μ ∈ M (Ω)

and η ∈ M (Γ) still belongs to W 1,p(Ω)∗. Since, in the case p > n, it holds thatp∗ = p# = +∞, we, for the sake of simplicity, have considered (and will consider)only those measures μ and η which are absolutely continuous19 in the presentedtext, except Sect. 3.2.5 below.

Convention 2.41 (Coercivity and a-priori estimates). The coercivity estimate(2.95) is just the so-called basic a-priori estimate, obtained by the test by so-lution u itself. Contrary to (2.95), it is routine to organize the terms having apositive sign in the left-hand side (and to estimate them from below typically byPoincare-type inequalities) while the other terms are put on the right-hand side(and to estimate them from above, e.g., by Holder and Young inequalities).

17Indeed, the mere monotonicity in the main part, i.e. (2.54), and (2.55), (2.57), (2.58), and(2.65), cannot be sufficient for the pseudomonotonicity of A. The counterexample is as follows:take c(x, r, s) ≡ c(s) with some c : Rn → R nonlinear, i.e. ∃s1, s2 ∈ Rn : 1

2c(s1) +

12c(s2) �=

c( 12s1 +

12s2) and take a = 0 at least on the line segment [s1, s2]. Then take a sequence {uk}k∈N

such that ∇uk is faster and faster oscillating between s1 and s2 (cf. Figure 3) on p.20 anduk(x)→ ( 1

2s1 + 1

2s2) · x.

18Indeed, the mapping u �→ c(u,∇u) is not totally continuous because it need not map weaklyconvergent sequences on strongly convergent ones. An example is as follows: take uk with anoscillating gradient if k odd and affine uk(x) = ( 1

2s1 + 1

2s2) · x if k even, so that again {uk}k∈N

converges weakly to this affine function but {c(∇uk)}k∈N does not converge at all if, e.g., c(s) =|2s− s1 − s2|.

19Those measures are known to have densities g ∈ L1(Ω) and h ∈ L1(Γ), respectively.


2.4.4 Higher-order equations

The generalization of the 2nd-order equation to equations involving 2k-orderderivatives, k ≥ 2, is often desirable. The corresponding boundary-value prob-lems then involve k-boundary conditions, called either the Dirichlet one if theyinvolve only derivatives up to (k−1)-order or the Neumann or the Newton one ifthey involve also derivatives of the order between k and 2k−1. We present herebriefly only quasilinear equations of the 4th order in a special20 divergence form

div(div(a(x, u,∇u,∇2u)

))+ c(x, u,∇u,∇2u) = g (2.97)

in Ω, with a : Ω×R×Rn ×Rn×n → Rn×n and c : Ω×R×Rn ×Rn×n → R. Here∇2u :=

[∂2

∂xi∂xju]ni,j=1

. More in detail, (2.97) means

n∑i,j=1

∂2

∂xi∂xjaij(x, u,∇u,∇2u

)+ c(x, u,∇u,∇2u

)= g. (2.98)

Formulation of natural boundary conditions is more difficult than for the 2nd-ordercase. The weak formulation is created by multiplying (2.97) by a test function v,by integration over Ω, and by using Green’s formula twice. Like in (2.50), thisgives ∫

Ω

a(x, u,∇u,∇2u) :∇2v +(c(x, u,∇u,∇2u)− g

)v dx

=

∫Γ

a(x, u,∇u,∇2u) : (ν ⊗∇v)− div(a(x, u,∇u,∇2u)

)·ν v dS. (2.99)

From this we can see that we must now cope with two boundary terms. In viewof this, the Dirichlet boundary conditions look as

u|Γ = uD and∂u

∂ν

∣∣∣Γ= u′D on Γ (2.100)

with uD and u′Dgiven. The weak formulation then naturally works with v∈V :=

W 2,p0 (Ω) = {v ∈ W 2,p(Ω); v|Γ = ∂v

∂ν |Γ = 0} with p > 1 an exponent relatedto qualification of the highest-order nonlinearity a(x, r, s, ·). This choice makesboth boundary terms in (2.99) zero; note that v|Γ = 0 makes also the tangentialderivative of v zero at a.a. x∈Γ hence ∂v

∂ν |Γ = 0 yields ∇v(x) = 0 on Γ.

By this argument, v|Γ = 0 makes ∇v = ∂v∂ν ν on Γ and allows us to write

a(x, u,∇u,∇2u):(ν⊗∇v)=a(x, u,∇u,∇2u):(ν⊗∂v

∂νν)=(νa(x, u,∇u,∇2u)ν

)∂v∂ν

and suggests that we formulate Dirichlet/Newton boundary conditions as

u|Γ = uD and νa(x, u,∇u,∇2u) ν + b(x, u,∇u) = h on Γ (2.101)

20See Exercises 2.98 and 4.32 for a more general case.


with uD and h given and b : Γ × R × Rn → R. This choice with v|Γ = 0 convertsthe boundary terms in (2.99) to

∫Γ(h− b(x, u,∇u)) ∂v∂νdS, which turns (2.99) just

into the integral identity∫Ω

a(x, u,∇u,∇2u) :∇2v + c(x, u,∇u,∇2u)v dx

+

∫Γ

b(x, u,∇u)∂v

∂νdS =

∫Ω

gv dx+

∫Γ

h∂v

∂νdS (2.102)

forming the weak formulation provided the test-function space V is taken as{v∈W 2,p(Ω); v|Γ = 0}.

If v|Γ is not fixed to zero, one must use a general decomposition ∇v =∂v∂ν ν + ∇Sv on Γ with ∇Sv = ∇v − ∂v

∂ν ν being the tangential gradient of v. Ona smooth boundary Γ, one can use another (now (n−1)-dimensional) Green-typeformula along the tangential spaces:21∫

Γ

a(x, u,∇u,∇2u):(ν⊗∇v) dS

=

∫Γ

(νa(x, u,∇u,∇2u)ν

)∂v∂ν

+ a(x, u,∇u,∇2u):(ν⊗∇Sv) dS

=

∫Γ


)∂v∂ν

− divS

(a(x, u,∇u,∇2u)ν

)v

+(divSν

)(νa(x, u,∇u,∇2u)ν

)v dS (2.103)

where divS := Tr(∇S) with Tr(·) being the trace of a (n−1)×(n−1)-matrix denotesthe (n−1)-dimensional surface divergence so that divSν is (up to a factor − 1

2 ) themean curvature of the surface Γ. Substituting it into (2.99), one obtains∫

Ω

a(x, u,∇u,∇2u) :∇2v +(c(x, u,∇u,∇2u)− g

)v dx

=

∫Γ


)∂v∂ν

−(div(a(x, u,∇u,∇2u)

)·ν+ divS

(a(x, u,∇u,∇2u)ν

)− (divSν)(νa(x, u,∇u,∇2u)ν

))v dS. (2.104)

This allows us to cast a natural higher-order Dirichlet/Newton boundary condition:

∂u

∂ν

∣∣∣Γ= u′

Dand (2.105a)

div(a(x, u,∇u,∇2u)

)·ν + divS

(a(x, u,∇u,∇2u)ν

)− (divSν

)(νa(x, u,∇u,∇2u)ν

)+ b(x, u,∇u) = h on Γ. (2.105b)

21This “surface” Green-type formula reads as∫Γ w:((∇Sv)⊗ν) dS =

∫Γ(divSν)(w:(ν⊗ν))v −

divS(w·ν)v dS. In the vectorial variant, this is used in mechanics of complex (also called nonsim-ple) continua, cf. [153, 337, 407]. For even 2k-order problems with k > 2 see also [244].


The underlying Banach space is then considered as V = {v∈W 2,p(Ω); ∂v∂ν = 0}

and the weak formulation is based on the integral identity:∫Ω

a(x, u,∇u,∇2u) :∇2v + c(x, u,∇u,∇2u)v dx

+

∫Γ

b(x, u,∇u)v dS =

∫Ω

gv dx+

∫Γ

hv dS. (2.106)

Eventually, the formula (2.104) reveals also the natural form of Newton-typeboundary conditions:

div(a(x, u,∇u,∇2u)

)·ν + divS

(a(x, u,∇u,∇2u)ν

)− (divSν

)(νa(x, u,∇u,∇2u)ν

)+ b0(x, u,∇u) = h0 and (2.107a)

νa(x, u,∇u,∇2u) ν + b1(x, u,∇u) = h1 on Γ. (2.107b)

The underlying Banach space can then be considered as V = W 2,p(Ω). The result-ing weak formulation of the boundary-value problem (2.97)–(2.107) then employsthe integral identity:∫

Ω

a(x, u,∇u,∇2u) :∇2v + c(x, u,∇u,∇2u)v dx

+

∫Γ

b0(x, u,∇u)v + b1(x, u,∇u)∂v

∂νdS =

∫Ω

gv dx+

∫Γ

h0v + h1∂v

∂νdS. (2.108)

For nonsmooth boundaries, these arguments based on formula (2.104) areno longer valid however and additional boundary terms can be seen; cf. [338] forboundaries with edges.

We will modify the Leray-Lions’ Theorem 2.36 for the case of the Dirichletconditions (2.100). Let us write naturally22 p∗∗ := (p∗)∗ and p∗# := (p∗)#. Forsimplicity, the assumptions are not the most general in the following assertion,whose proof, paraphrasing that of Theorem 2.36, is omitted here.

Proposition 2.42 (Existence for Dirichlet problem). Let a(x, r, s, ·) :Rn×n → Rn×n be strictly monotone,

∃k∈L1(Ω), 1<q≤ p : a(x, r, s, S):S+c(x, r, s, S)r ≥ ε|S|p+ε|r|q−k(x), (2.109a)

∃γ∈Lp′(Ω) : |a(x, r, s, S)| ≤ γ(x) + C|r|(p∗∗−ε)/p′

+ C|s|(p∗−ε)/p′+ C|S|p−1, (2.109b)

∃γ∈Lp∗∗′+ε(Ω) : |c(x, r, s, S)| ≤ γ(x) + C|r|p∗∗−ε−1

+ C|s|(p∗−ε)/p∗∗′+ C|S|(p−ε)/p∗∗′

, (2.109c)

22This means p∗∗ = np/(n−2p) if p < n/2 or p∗∗ < +∞ if p = n/2 or p∗∗ = +∞ if p > n/2,cf. Corollary 1.22 for k = 2. For p∗# = (np−p)/(n−2p) if p < n/2, cf. Exercise 2.70.


with some C ∈R+ and ε, ε > 0 and again the Convention 2.26 (now concerningp∗∗ = +∞ for p > n/2), and let uD = v|Γ and u′

D= ∂v

∂ν for some v ∈W 2,p(Ω),

and g∈Lp∗∗′(Ω). Then the boundary-value problem (2.97) with (2.100) has a weak

solution, i.e. (2.99) holds for all v ∈ W 2,p0 (Ω) together with the boundary conditions

(2.97).

For the Newton boundary conditions (2.107), the analog of the existenceassertion looks as follows:

Proposition 2.43 (Existence for Newton problem). Let a, c, and g be as inProposition 2.42 and satisfy (2.109), and let b0 and b1 satisfy

∃k∈L1(Γ) : b0(x, r, s)r + b1(x, r, s)(s·ν(x)) ≥ −k(x), (2.110a)

∃γ∈Lp∗#′(Γ) :

∣∣b0(x, r, s)∣∣ ≤ γ(x) + C|r|p∗#−ε−1+ C|s|(p#−ε)/p∗#, (2.110b)

∃γ∈Lp#′(Γ) :

∣∣b1(x, r, s)∣∣ ≤ γ(x) + C|r|(p∗#−ε)/p#′+ C|s|p#−ε−1 (2.110c)

with some C ∈R+ and ε > 0, and let h0 ∈ Lp∗#′(Γ) and h1 ∈ Lp#′

(Γ). Then theboundary-value problem (2.97) with (2.107) has a weak solution, i.e. (2.108) holdsfor all v ∈ W 2,p

0 (Ω).

The modification for other boundary conditions (2.101) or (2.105) can easilybe cast and is left as an exercise.

As pointed out before, one should care about consistency and selectivityof the definitions of weak solutions. Consistency is guaranteed by the derivationof the weak formulation itself. Let us illustrate the selectivity, i.e. an analog ofProposition 2.25, on the most complicated case of the Newton boundary-valueproblem:

Proposition 2.44 (Selectivity of the weak-solution definition). Let Γbe smooth, a ∈ C2(Ω×R×Rn×Rn×n;Rn×n), c ∈ C0(Ω×R×Rn×Rn×n), andb0, b1 ∈ C0(Γ×R×Rn), g ∈ C(Ω), and h0, h1 ∈ C(Γ). Then any weak solutionu ∈ C4(Ω) of the boundary-value problem (2.97) with (2.107) is the also classicalsolution.

Proof. Put v ∈ V = W 2,p(Ω) into (2.108) and use Green’s formula (1.54) twice,as well as the surface Green formula (2.103). One gets∫

Ω

(div2a(x, u,∇u) + c(x, u,∇u)− g

)v dx

+

∫Γ

(div(a(x, u,∇u,∇2u)

)·ν + divS

(a(x, u,∇u,∇2u)ν

)− (divSν

)(νa(x, u,∇u,∇2u)ν

)+ b0(x, u,∇u)− h0

)v dS

+

∫Γ

(νa(x, u,∇u,∇2u) ν + b1(x, u,∇u)− h1

)∂v∂ν

dS = 0. (2.111)


Considering v with a compact support in Ω, one has v|Γ = 0 = ∂v∂ν and both

boundary integrals in (2.111) vanish. As v is otherwise arbitrary, one deduces that(2.97) holds a.e., and hence even everywhere in Ω due to the assumed smoothnessof a and c. Hence, the first integral in (2.111) vanishes. Then, put a more generalv∈V into (2.111) but still such that ∂v

∂ν = 0. Thus the second boundary integralin (2.111) vanishes. From the first boundary integral, we recover the boundarycondition (2.107a).23 Due to the assumed smoothness of a and continuity of b0 andh0, (2.107a) holds pointwise. Finally, we can take v ∈ V fully general. Knowingalready that the first and the second integral in (2.111) vanish, from the lastintegral we can recover the remaining boundary condition (2.107b).24 �

Remark 2.45 (Other boundary conditions). The above four combinations ofboundary conditions still do not represent the whole class of variationally con-sistent boundary conditions for equation (2.97). For α0, α1 ∈ L∞(Γ), one canconsider a combined condition composed from (2.101) and (2.105), namely

α1∂u

∂ν+ α0u = uD and (2.112a)

α1

(div(a(x, u,∇u,∇2u)

)·ν + divS

(a(x, u,∇u,∇2u)ν

)− (divSν

)(νa(x, u,∇u,∇2u)ν

))+ α0ν

a(x, u,∇u,∇2u) ν + b(x, u,∇u) = h on Γ. (2.112b)

The underlying Banach space is then V = {v ∈ W 2,p(Ω); α1∂u∂ν + α0u = 0} and

the weak formulation is again (2.106) with b = b2/α1 and h = h2/α1 providedα1 �= 0. Alternatively, for α0 �= 0 one can rather pursue the weak formulationbased on (2.102).

Example 2.46 (p-biharmonic operator). A concrete choice of a from (2.97)

aij(x, r, s, S) :=

{ ∣∣∑nk=1 Skk

∣∣p−2∑nk=1 Skk for i = j,

0 for i �= j,(2.113)

converts div div a(x, u,∇u,∇2u) into the so-called p-biharmonic operatorΔ(|Δu|p−2Δu). Applying Green’s formula twice to this operator tested by v yieldsthe identity

23Here the important fact is that the set {v|Γ; v ∈ V, ∂v∂ν

= 0} is still dense in L1(Γ). Indeed,

any v ∈ W 1,2(Ω) can be modified to uε so that (vε−v)|Γ is small but ∂∂ν

vε = 0 on Γ. To outlinethis procedure, first we rectify Γ locally so that we can consider a half-space, cf. Fig. 8 on p. 91below, then extend v by reflection of v with respect to Γ, and eventually mollify the extended v.

24Here the important fact is that the set { ∂v∂ν

; v ∈ V } is dense in L1(Γ), which can be seenby a local rectification of Γ and by an explicit construction of v in the vicinity of Γ with a givensmooth ∂v

∂νand, e.g., zero trace on Γ.

2.5. Weakly continuous mappings, semilinear equations 61∫Ω

Δ(|Δu|p−2Δu

)v dx

= −∫Ω

∇(|Δu|p−2Δu)·∇v dx+

∫Γ

∂

∂ν

(|Δu|p−2Δu)v dS

=

∫Ω

|Δu|p−2ΔuΔv dx+

∫Γ

∂

∂ν

(|Δu|p−2Δu)v − |Δu|p−2Δu

∂v

∂νdS, (2.114)

from which, besides the Dirichlet conditions (2.100), one can pose naturally alsoDirichlet/Newton conditions (2.101) now in the form

u|Γ = uD and |Δu|p−2Δu+ b(x, u,∇u) = h, (2.115)

or the higher Dirichlet/Newton conditions (2.105) now in the simpler form

∂u

∂ν

∣∣∣Γ= u′

Dand

∂

∂ν

(|Δu|p−2Δu)+ b(x, u,∇u) = h, (2.116)

or also the Newton condition (2.107) now in the simpler form

∂

∂ν

(|Δu|p−2Δu)+ b0(x, u,∇u) = h0, |Δu|p−2Δu + b1(x, u,∇u) = h1. (2.117)

Note that (2.116) and (2.117) do not contain the divS-terms because, in-stead of ν⊗∇v in (2.99), one has ν·∇v = ∂v

∂ν in (2.114). The pointwise co-ercivity (2.109a) cannot be satisfied for (2.113), however, and the coerciv-ity of A on V must rely on a delicate interplay with the boundary con-ditions. E.g., for Dirichlet conditions (2.100) with uD = 0 = u′D and forp = 2, one has by using Green’s formula twice 〈A(u), u〉 =

∫Ω|Δu|2dx =

− ∫Ω∇u·∇Δudx = − ∫Ω∇u·div(∇2u) dx =∫Ω |∇2u|2dx, which thus controls ∇2u

in L2(Ω;Rn×n). Another example is the Newton’s condition (2.117) withb0(x, r, s) = β0(x)r, b1(x, r, s) = −β1(x)(s·ν), and p = 2, one has 〈A(u), u〉 =∫Ω |Δu|2dx +

∫Γ β0u

2 + β1(∂∂νu)

2dS. This is a continuous quadratic form onW 2,2(Ω) and for the Poincare-like inequality 〈A(u), u〉 ≥ C

P‖u‖2W 2,2(Ω) it suf-

fices to guarantee that 〈A(u), u〉 = 0 implies u = 0. This can be done by assum-ing β0, β1 ≥ 0, and β0 or β1 positive on a “sufficiently large part” of Γ.25

2.5 Weakly continuous mappings, semilinear equations

In case that A is coercive and, instead of being pseudomonotone, is weakly contin-uous, we can prove existence of a solution to A(u) = f much more easily. Althoughthe assumption of the weak continuity is restrictive, such mappings enjoy still aconsiderably large application area. Here, we can even advantageously generalizethe concept for mappings A : V → Z∗ for some Banach space Z ⊂ V densely sothat Z∗ ⊃ V ∗. If Vk ⊂ Z for any k ∈ N, we can modify (2.5) and then Theorem 2.6:

25Here, a certain caution is advisable: e.g. for Ω a square [0, 1]2, it is not sufficient if β0(·) = 1on the sides with x1 = 0 and x2 = 0 and otherwise β0 and β1 vanishes because of existence of anon-vanishing function u(x) = x1x2 for which 〈A(u), u〉 = 0.


Proposition 2.47 (Existence). If a weakly continuous mapping A : V → Z∗ iscoercive in the modified sense

lim‖v‖V→∞

v∈Z

〈A(v), v〉Z∗×Z

‖v‖V = +∞, (2.118)

and if f ∈ V ∗, then the equation A(u) = f has a solution.

Proof. The technique of the proof of Theorem 2.6 allows for a very simple mod-ification: instead of (2.14), we consider the Galerkin identity (2.8) as 〈A(uk) −f, vk〉Z∗×Z = 0 for vk ∈ Vk such that vk → v in Z, and make a direct limitpassage. Note that (2.13) looks now as

ζ(‖uk‖V

)∥∥uk

∥∥V≤ ⟨A(uk), uk

⟩Z∗×Z

=⟨f, uk

⟩Z∗×Z

=⟨f, uk

⟩V ∗×V

≤ ∥∥f∥∥V ∗∥∥uk

∥∥V

and again yields {uk}k∈N bounded in V because f ∈ V ∗. �Confining ourselves again to the 2nd-order problems as in Sections 2.4.1–

2.4.3, we can easily use this concept for the special case when a(x, r, ·) : Rn → Rn

and c(x, r, ·) : Rn → R are affine, we will call such problems as semilinear althoughsometimes this adjective needs still a(x, ·, s) constant as in (0.1). So, here

ai(x, r, s) :=n∑

j=1

aij(x, r)sj + ai0(x, r), i = 1, . . . , n, (2.119a)

c(x, r, s) :=n∑

j=1

cj(x, r)sj + c0(x, r), (2.119b)

with aij , cj : Ω × R → R Caratheodory mappings whose growth is now to bedesigned to induce the Nemytskiı mappings N(ai1,...,ain),N(c1,...,cn) : L

2∗−ε(Ω) →L2(Ω;Rn) and Nai0 ,Nc0 : L2∗−ε(Ω) → L1(Ω) with ε > 0. Besides, the bound-ary nonlinearity b : Γ × R → R is now to induce the Nemytskiı mapping

Nb : L2#−ε(Γ) → L1(Γ). This means, for i, j = 1, . . . , n,

∃γ1 ∈ L2(Ω), C ∈ R : |aij(x, r)| ≤ γ1(x) + C|r|(2∗−ε)/2,

|cj(x, r)| ≤ γ1(x) + C|r|(2∗−ε)/2, (2.120a)

∃γ2 ∈ L1(Ω), C ∈ R : |ai0(x, r)| ≤ γ2(x) + C|r|2∗−ε,

|c0(x, r)| ≤ γ2(x) + C|r|2∗−ε, (2.120b)

∃γ3∈L1(Γ), C ∈ R : |b(x, r)| ≤ γ3(x) + C|r|2#−ε. (2.120c)

The exponent p = 2 is natural because a(x, r, ·) has now a linear growth. Note thatthese requirements just guarantee that all integrals in (2.51) have a good sense ifv ∈ W 1,∞(Ω) =: Z. Again, Convention 2.26 on p. 46 is considered.

2.5. Weakly continuous mappings, semilinear equations 63

Lemma 2.48 (Weak continuity of A). Let (2.119)–(2.120) hold. Then A isweakly* continuous as a mapping W 1,2(Ω) → W 1,∞(Ω)∗.

Proof. Having a weakly convergent sequence {uk}k∈N in W 1,2(Ω), this sequenceconverges strongly in L2∗−ε(Ω). Then, by the continuity of the Nemytskiı map-pings N(ai1,...,ain),N(c1,...,cn) : L

2∗−ε(Ω) → L2(Ω;Rn) and Nai0 ,Nc0 : L2∗−ε(Ω) →L1(Ω), it holds that

limk→∞

∫Ω

n∑i=1

( n∑j=1

aij(uk)∂uk

∂xj+ ai0(uk)

) ∂v

∂xi+( n∑

j=1

cj(uk)∂uk

∂xj+ c0(uk)

)v dx

=

∫Ω

n∑i=1

( n∑j=1

aij(u)∂u

∂xj+ ai0(u)

) ∂v

∂xi+( n∑

j=1

cj(u)∂u

∂xj+ c0(uk)

)v dx

for k → ∞ and any v ∈ W 1,∞(Ω). Also uk|Γ → u|Γ in L2#−ε(Γ), and, by (2.120c),we have convergence in the boundary term

∫Γb(uk)v dS → ∫

Γb(u)v dS. �

Proposition 2.49 (Existence of weak solutions). Let (2.119)–(2.120) hold,

g∈L2∗′(Ω), h∈L2#

′(Γ), and, for some ε> 0, γ1 ∈L2(Ω), γ2 ∈L1(Ω), γ3 ∈L1(Γ),

and for a.a. x∈Ω (resp. x∈Γ for (2.121b)) and all (r, s)∈R1+n, it holds that

n∑i=1

( n∑j=1

aij(x, r)sj + ai0(x, r))si +

( n∑j=1

cj(x, r)sj + c0(x, r))r

≥ ε|s|2 + ε|r|2 − γ1(x)|s| − γ2(x), (2.121a)

b(x, r)r ≥ −γ3(x). (2.121b)

Then the boundary-value problem (2.45) with (2.49) has a weak solution in thesense of Definition 2.24 using now v ∈ W 1,∞(Ω).

Proof. We can use the abstract Proposition 2.47 now with V := W 1,2(Ω), Z :=W 1,∞(Ω), and Vk some finite-dimensional subspaces ofW 1,∞(Ω) satisfying (2.7).26

The coercivity (2.118) is implied by (2.121) by routine calculations.27 Then we useLemma 2.48 and Proposition 2.47. �

Remark 2.50 (Conventional weak solutions). Let, in addition to the assumptionsof Proposition 2.49, also the growth condition (2.55) with p = 2 hold. Then thesolution obtained in Proposition 2.49 allows for v ∈ W 1,2(Ω) in Definition 2.24.

26Such subspaces always exists, e.g. one can imagine subspaces as in Example 2.67.27We have 〈A(v), v〉= ∫

Ω

∑ni=1

(∑nj=1 aij(v)

∂∂xj

v+ai0(v))

∂∂xi

v+(∑n

j=1 cj(v)∂

∂xjv+c0(v)

)v

dx+∫Γb(v)v dS ≥ ε‖∇v‖2

L2(Ω;Rn)− 1

2ε‖γ1‖2L2(Ω)

− ε2‖∇v‖2

L2(Ω;Rn)− ‖γ2‖L1(Ω) − γ3‖v‖1L1(Γ)

.


2.6 Examples and exercises

This section contains both exercises to make the above presented theory more com-plete and some examples of analysis of concrete semi- and quasi-linear equations.The exercises will mostly be accompanied by brief hints in the footnotes.

2.6.1 General tools

Exercise 2.51 (Banach’s selection principle). Assuming the sequential compactnessof closed bounded intervals in R is known, prove Banach’s Theorem 1.7 by asuitable diagonalization procedure.28

Exercise 2.52 (Uniform convexity of Hilbert spaces). For V being a Hilbert space,prove the assertion of Theorem 1.2 directly.29 Using (1.4), prove that any Hilbertspace is uniformly convex.30

Exercise 2.53 (Pseudomonotonicity). Assuming (2.3a), show that (2.3b) is equiv-alent to31

uk ⇀ u,

lim supk→∞

⟨A(uk), uk−u

⟩ ≤ 0

}⇒

{w-lim

k→∞A(uk) = A(u),

limk→∞

⟨A(uk), uk

⟩=⟨A(u), u

⟩.

(2.122)

Exercise 2.54 (Weakening of pseudomonotonicity). Modify the proof of BrezisTheorem 2.6 for A coercive, bounded, demicontinuous, and satisfying32

uk ⇀ u & A(uk) ⇀ flim supk→∞

⟨A(uk), uk

⟩ ≤ 〈f, u〉}

⇒ f = A(u). (2.123)

28Hint: Consider a sequence {fk}k∈N bounded in V ∗ and a countable dense subset {vk}k∈N

in V , take v1 and select an infinite subset A1 ⊂ N such that the sequence of real numbers{〈fk, v1〉}k∈A1

converges in R to some f(v1), then take v2 and select an infinite subset A2 ⊂ A1

such that {〈fk, v2〉}k∈A2converges to some f(v2), etc. for v3, v4, . . . . Then make a diagonaliza-

tion procedure by taking lk the first number in Ak which is greater than k. Then {〈flk , vi〉}k∈N

converge to f(vi) for all i ∈ N. Show that f is linear on span({vi}i∈N) and bounded because|f(vi)| ≤ limk→∞ |〈flk , vi〉| ≤ lim supk→∞ ||fk||∗||vi||, and finally extend f on the whole V ∗ justby continuity.

29Hint: ‖uk‖→‖u‖ and uk⇀u imply ‖uk−u‖2 = ‖uk‖2 +(u−2uk, u)→ ‖u‖2 +(u−2u, u) = 0.30Hint: Realize that ‖u‖ = 1 = ‖v‖ and ‖u − v‖ ≥ ε in (1.5) imply 1

2‖u + v‖ =√

(u, v) − ‖u−v‖2/4 ≤ √1− ε2/4 ≤ 1 − δ provided 0 < δ ≤ 1 solves δ2 − 2δ + ε2/4 = 0.

Such δ exists if 0 < ε ≤ 2, while for ε > 2 the implication (1.5) is trivial.31Hint: (2.122)⇒(2.3b) is trivial. The converse implication: by (2.3a), assume A(uk) ⇀ f (a

subsequence), then 0 ≥ lim supk→∞〈A(uk), uk−u〉 = limsupk→∞〈A(uk), uk〉 − 〈f, u〉 implies〈A(u), u − v〉 ≤ lim infk→∞〈A(uk), uk−v〉 ≤ lim supk→∞〈A(uk), uk〉 − 〈f, v〉 ≤ 〈f, u−v〉, fromwhich A(u) = f , hence A(uk) ⇀ f (the whole sequence), and eventually (2.3b) for v = 0 yields

〈A(u), u〉 ≤ lim infk→∞

〈A(uk), uk〉 ≤ lim supk→∞

〈A(uk), uk〉 ≤ limk→∞

〈A(uk), u〉 = 〈A(u), u〉.

32Hint: Modify Step 4 of the proof of Theorem 2.6: as both {uk}k∈N and A are bounded,A(uk) ⇀ χ (as a subsequence) and, from (2.8), χ = f , hence A(uk) ⇀ f (the whole sequence)and, again by (2.8), 〈A(uk), uk〉 = 〈f, uk〉 → 〈f, u〉. Then by (2.123) f = A(u).

2.6. Examples and exercises 65

Show that any pseudomonotone A satisfies (2.123).33

Exercise 2.55 (Tikhonov-type modification34 of Schauder’s Theorem 1.9). Assum-ing a reflexive separable Banach space V � V1, show that a weakly continuousmapping M : V → V which maps a ball B in V into itself has a fixed point.35

Exercise 2.56 (Direct method for A weakly continuous). Assume A : V → V ∗

weakly continuous, V Hilbert, and modify the Brezis Theorem 2.6 by using directlySchauder fixed-point Theorem 1.9 without approximating the problem.36

Exercise 2.57. Try to make a limit passage in (2.38)–(2.39) simultaneously in iand l by considering i = l. Realize why it was necessary to make the double limitliml→∞ limi→∞ instead of liml=i→∞ in the proof of Proposition 2.17.

Exercise 2.58. Assuming 1 ≤ q ≤ p < +∞, evaluate the norms of the continuousembeddings L∞(Ω) ⊂ Lp(Ω) ⊂ Lq(Ω).37

Exercise 2.59 (Interpolation of Lebesgue spaces). Prove (1.23) by using Holder’sinequality.38

Exercise 2.60 (Continuity of Nemytskiı mappings). Show that the Nemytskiı map-ping Na with a satisfying (1.48) is a bounded continuous mapping Lp1(Ω) ×

33Hint: The premise of (2.123) and the pseudomonotonicity implies lim supk→∞〈A(uk), uk −u〉 = lim supk→∞〈A(uk), uk〉 − limk→∞〈A(uk), u〉 ≤ 〈f, u〉 − 〈f, u〉 = 0 so that, by (2.3b),〈A(u), u − v〉 ≤ lim infk→∞〈A(uk), uk − v〉 ≤ lim supk→∞〈A(uk), uk − v〉 ≤ 〈f, u − v〉 for anyv ∈ V , from which f = A(u) indeed follows.

34Tikhonov [413] proved a bit more general assertion, known now as Tikhonov’s theorem: acontinuous mapping from a compact subset of a locally convex space into itself has a fixed point.

35Hint: Consider B endowed with a weak topology, realize that uk → u in V1 and uk ∈ Bimplies uk ⇀ u in B, hence M(uk) ⇀ M(u) in V and then also M(uk)→M(u) in V1, and thenuse Schauder’s Theorem 1.9.

36Hint: Repeat Step 2 of the proof of Brezis Theorem 2.6 directly for V instead of Vk . Usethe weak topology on {v ∈ V ; ‖v‖ ≤ �}, and realize that Ik is to be omitted while J−1

k is tobe weakly continuous (which really is due to its demicontinuity, cf. Corollary 3.3 below, and itslinearity, cf. Remark 3.10). Also use Exercise 2.55.

37Hint: Estimate∥∥u∥∥Lp(Ω)

= p√∫

Ω

∣∣u∣∣pdx ≤ p

√∫Ω ess sup

ξ∈Ω

∣∣u(ξ)∣∣pdx = p

√∥∥u∥∥pL∞(Ω)

∫Ω 1dx = N

∥∥u∥∥L∞(Ω)

with N =(measn(Ω)

)1/pbeing the norm of the embedding L∞(Ω) ⊂ Lp(Ω). Likewise, by

Holder’s inequality,

‖u‖qLq(Ω)

=

∫Ω1 · ∣∣u∣∣qdx ≤ (p/q)′

√∫Ω1dx p/q

√∫Ω

∣∣u∣∣pdx = Nq∥∥u∥∥q

Lp(Ω)

with N =(measn(Ω)

)(p−q)/(pq).

38Hint: Use Holder’s inequality for∫Ω|v|pdx =

∫Ω|v|λp|v|(1−λ)pdx ≤

(∫Ω|v|λpαdx

)1/α(∫Ω|v|(1−λ)pβdx

)1/β

with a suitable α = p1/(λp) and β = p2/((1 − λ)p), namely α−1 + β−1 = 1 which just meansthat p satisfies the premise in (1.23).


Lp2(Ω;Rn) → Lp0(Ω).39

Exercise 2.61. Show that p3 in Theorem 1.27 indeed cannot be +∞: find some asatisfying (1.48) for p1, p2 < +∞ and p3 = +∞ such that Na is not continuous.40

Exercise 2.62. Show that, for any c : Rm1 → Rm2 not affine, the Nemytskiımapping Nc : u �→ c(u) is not weakly continuous; modify Figure 3 on p.20.41

Exercise 2.63 (Integral balance (2.63)). Consider the test volume in the integralbalance (2.63) as a ball O = {x; |x − x0| ≤ �} and derive (2.63) for a.a. � bya limit passage in the weak formulation (2.45) tested by v = vε with vε(x) :=(1− dist(x,O)/ε)+ provided the basic data qualification (2.55a,c) is fulfilled.42

Exercise 2.64. Show that the mapping u �→ c(u,∇u) is compact, i.e. it mapsbounded sets inW 1,p(Ω) into relatively compact sets inW 1,p(Ω)∗, cf. Remark 2.39.For this, specify a growth assumption on c.43

Exercise 2.65. By using (2.56), show that A : W 1,p(Ω) → W 1,p(Ω)∗ defined by(2.59) is demicontinuous. Note that no monotonicity of this A is needed, contraryto an abstract case addressed in Lemma 2.16.

39Hint: Take uk → u in Lp1(Ω) and yk → y in Lp2(Ω;Rn), then take subsequences converginga.e. on Ω. Then, by continuity of a(x, ·, ·) for a.a. x ∈ Ω, Na(uk, yk) → Na(u, y) a.e., and byProposition 1.13(i), in measure, too. Due to the obvious estimate

∣∣a(x, uk, yk)− a(x, u, y)∣∣p0 ≤ 6p0−1

(γp0 (x) +C

∣∣uk(x)∣∣p1+C

∣∣u(x)∣∣p1+C∣∣yk(x)∣∣p2+C

∣∣y(x)∣∣p2)for a.a. x ∈ Ω, show that {|a(x, uk, yk) − a(x, u, y)|p0}k∈N is equi-absolutely continuous sincestrongly convergent sequences are; use e.g. Theorem 1.16(i)⇒(iii). Eventually combine thesetwo facts to get

∫Ω |a(x, uk, yk) − a(x, u, y)|p0dx → 0 and realize that, as the limit Na(u, y) is

determined uniquely, eventually the whole sequence converges.40Hint: For example, a(x, r, s) = r/(1 + |r|) and uk = χAk

, a characteristic function of a set

Ak, measn(Ak) > 0, limk→∞measn(Ak) = 0, and realize that ‖uk‖Lp(Ω) =(measn(Ak))1/p → 0

but ‖Na(uk)‖L∞(Ω) = 1/2 �→ 0 = ‖Na(0)‖L∞(Ω).41Hint: Take r1, r2 ∈ Rm1 such that c( 1

2r1+

12r2) �= 1

2c(r1)+

12c(r1) and a sequence of functions

oscillating faster and faster between r1 and r2 (instead of 1 and −1 as used on Figure 3).42Hint: Putting x0 = 0 without any loss of generality, realizing that ∇vε(x) = −ε−1x/|x| if

� < |x| < �+ ε otherwise ∇vε(x) = 0 a.e. and that ν(x) = x/|x|, the limit passage

0 =

∫|x|≤

c(x, u,∇u)− g(x) dx+

∫≤|x|≤+ε

((c(x, u,∇u)− g(x)

)(1− |x|−�

ε

)

− 1

εa(x, u,∇u) · x

|x|)dx→

∫|x|≤

c(x, u,∇u)− g(x) dx−∫|x|=

a(x, u,∇u) · ν dS

holds at every right Lebesgue point of the function f : � �→ �−1∫|x|= a(x, u,∇u) · xdS, i.e. at

every � such that f(�) = limε↘01ε

∫ +ε f(ξ)dξ. As f is locally integrable thanks to the growth

conditions (2.55a,c), it is known that, for a.a. �, it enjoys this property.43Hint: It suffices to design the growth condition so that Nc maps Lp∗(Ω) × Lp(Ω;Rn) into

L(p∗−ε)′ (Ω) which is compactly embedded into W 1,p(Ω)∗.


Exercise 2.66 (V -coercivity). Consider, instead of (2.92),

∃ε1, k0>0, k1∈L1(Ω) : a(x, r, s)·s+c(x, r, s)r ≥ ε1|s|p−k0|r|q1−k1(x), (2.124a)

∃ε2 > 0, k2∈L1(Γ) : b(x, r)r ≥ ε2χΓN(x)|r|q−k2(x), (2.124b)

for some 1 < q1 < q ≤ p and measn−1(ΓN) > 0, and prove Lemma 2.35 by usingthe Poincare inequality in the form (1.56). Likewise, formulate similar conditionsfor the case of mixed Dirichlet/Newton conditions (2.49) and use (1.57) to showcoercivity of the shifted operator A0 = A(·+w) with w|ΓD = uD on V from (2.52).

Example 2.67 (Finite-element method). As an example for the finite-dimensionalspace Vk used in Galerkin’s method in the concrete case V = W 1,p(Ω), the readercan think of Tk as a simplicial partition of a polyhedral domain Ω ⊂ Rn, i.e. Tk

is a collection of n-dimensional simplexes having mutually disjoint interiors andcovering Ω; if n = 2 or 3, it means a triangulation or a “tetrahedralization” as onFigure 5a or 5b, respectively. Then, one can consider Vk := {v ∈W 1,p(Ω); ∀S ∈Tk : v|S is affine}. A canonical base of Vk is formed by “hat” functions vanishingat all mesh points except one; cf. Figure 5a.44 Nested triangulations, i.e. eachtriangulation Tk+1 is a refinement of Tk, obviously imply Vk ⊂ Vk+1 which wehave used in (2.7).

Ω

x1

x2

v = v(x1, x2)

Ω↖

Figure 5a. Triangulation of a polygo-nal domain Ω ⊂ R2 and oneof the piece-wise affine ‘hat-shaped’ base functions.

Figure 5b. A fine 3-dimensional tetrahedralmesh on a complicated (but stillsimply connected) Lipschitz do-main Ω⊂R3; courtesy of M.Madlık.

This is the so-called P1-finite-element method. Often, higher-order polynomialsare used for the base functions, sometimes in combination with non-simplecticmeshes. For non-polyhedral domains, one can use a rectification of the curvedboundary by a certain homeomorphism as on Figure 8 on p. 91. Efficient softwarepackages based on finite-element methods are commercially available, includingroutines for automatic mesh generation on complicated domains, as illustrated onFigure 5b.

Exercise 2.68. Assuming n = 1 and limk→∞maxS∈Tkdiam(S) = 0, prove density

of⋃

k∈N Vk in W 1,p(Ω), cf. (2.7), for Vk from Example 2.67.45

44In such a base, the local character of differential operators is reflected in the Galerkin schemethat, e.g., linear differential operators result in matrices which are sparse.

45Hint: By density Theorem 1.25, take v ∈ W 2,∞(Ω) and vk ∈ Vk such that vk(x) = v(x)


Remark 2.69. To ensure (2.7) if n ≥ 2, a qualification of the triangulation isnecessary; usually, for some ε > 0, one requires that always diam(S)/�S ≥ ε withdenoting �S the radius of a ball contained in S.

Exercise 2.70 (Traces of higher-order Sobolev spaces). Generalize the trace Theo-rem 1.23 forW 2,p(Ω), and identify the integrability exponent for traces of functionsfrom W 2,p(Ω), namely p∗# := (p∗)#, as (np−p)/(n−2p) if 2p < n, otherwise, itsintegrability is arbitrarily large if 2p = n or in L∞(Γ) if 2p > n. Continue byinduction for W k,p(Ω), k ≥ 3.46

2.6.2 Semilinear heat equation of type −div(A(x, u)∇u) = g

Here we focus on a heat equation where, from physical reasons, the heat-transfercoefficients depend typically on temperature but not on its gradient, giving riseto a semilinear equation as investigated in Section 2.5. Moreover, we speak abouta critical growth of the particular nonlinearity when (2.55) would be fulfilled onlyif ε = 0. Here we will meet the situation when even ε = −1 in (2.55b) is needed(and by replacing the conventional Sobolev space W 1,p(Ω) by (2.128) eventuallyallowed) for b(x, ·); this is reported as a super-critical growth.

Example 2.71 (Nonlinear heat equation). The steady-state heat transfer in a non-homogeneous anisotropic nonlinear47 medium with a boundary condition control-ling the heat flux through two mechanisms, convection and Stefan-Boltzmann-typeradiation48 as outlined on Figure 6a, is described by the following boundary-valueproblem

−div(A(x, u)∇u

)= g(x) on Ω,

νA(x, u)∇u = b1(x)(θ − u)︸︷︷︸convectiveheat flux

+ b2(x)(θ4 − |u|3u)︸︷︷︸

radiativeheat flux

on Γ,

}(2.125)

at every x ∈ Ω which is a mesh point of the partition Tk, and ‖∇vk − ∇v‖L∞(Ω;Rn) ≤diam(S)‖∇2v‖L∞(Ω;Rn×n); as n = 1, each S is an interval here.

46Hint: For W 2,p(Ω) with 2p < n, consider W 2,p(Ω) ⊂ W 1,p∗(Ω) and apply Theorem 1.23 forp∗ = np/(n−p) instead of p. By induction, u �→ u|Γ : W k,p(Ω)→ L(np−p)/(n−kp)(Γ) if kp < n.

47The adjective “nonhomogeneous” refers to spatial dependence of the material properties, hereA. The adjective “anisotropic” means that A �= I in general, i.e. the heat flux is not necessarilyparallel with the temperature gradient and applies typically in single-crystals or in materialswith a certain ordered structure, e.g. laminates. The adjective “nonlinear” is related here toa temperature dependence of A which applies especially when the temperature range is large.E.g. heat conductivity in conventional steel varies by tens of percents when temperature rangeshundreds degrees; cf. [358].

48Recall the Stefan-Boltzmann radiation law: the heat flux is proportional to u4−θ4 where θ isthe absolute temperature of the outer space. In room temperature, the convective heat transfer,proportional to u− θ through the coefficient b1, is usually dominant. Yet, for example, in steel-manufacturing processes the radiative heat flux becomes quickly dominant when temperaturerises, say, above 1000 K and definitely cannot be neglected; cf. [358].


withu = temperature in a thermally conductive body occupying Ω,θ = temperature of the environment,A = [aij ]

ni,j=1= a symmetric heat-conductivity matrix, A :Ω×R→Rn×n, i.e.

ai(x, r, s) =∑n

j=1 aij(x, r)sj ,

A(x, u)∇u =(∑n

j=1 aij(u)∂

∂xju)ni=1

=the heat flux,

νA(x, u)∇u =∑n

i,j=1 aij(u)νi∂

∂xju =the heat flux through the boundary,

b1, b2 = coefficients of convective and radiative heat transfer through Γ,g = volume heat source.

Ω Ω

RADIATIONRADIATION

CONVECTION

CONVECTION

HEAT FLUXINSIDE THE PLATE

Figure 6a. Illustration of a heat-trans-fer problem in a 3-dimen-sional body Ω ⊂ R3.

Figure 6b. Illustration of a heat-trans-fer problem in a 2-dimen-sional plate Ω ⊂ R2.

In the setting (2.48), b(x, r) = b1(x)r + b2(x)|r|3r and h(x) = [b1θ + b2θ4](x). We

assume θ ≥ 0, b1(x) ≥ b1 > 0 and b2(x) ≥ b2 > 0, b1 ∈ L5/3(Γ) and b2 ∈ L∞(Γ),and the operator A is defined by⟨

A(u), v⟩:=

∫Ω

(∇v)A(x, u)∇u dx+

∫Γ

(b1(x)u + b2(x)|u|3u

)v dS. (2.126)

It should be emphasized that no monotonicity of A with respect to the L2-innerproduct can be expected if A(x, ·) is not constant.49Exercise 2.72 (Pseudomonotone-operator approach). Check the assumptions(2.55) and (2.65) in Section 2.450 as well as the coercivity (2.124)51. Realize, in

49This means∫Ω(∇u1 −∇u2)(A(x, u1)∇u1 − A(x, u2)∇u2) dx < 0 may occur.

50Hint: The assumption (2.55a) reads here as |∑nj=1 aij(x, r)sj | ≤ γ(x) + C|r|(p∗−ε)/p′ +

C|s|p−1 with γ ∈ Lp′(Ω). This requires p ≥ 2. The assumption of monotonicity in the main part(2.65) just requires that A(x, r) = [aij(x, r)] is positive semi-definite for all r and a.a. x ∈ Ω,i.e. sA(x, r)s ≥ 0. The assumption (2.55b) for the “physical” dimension n = 3 and for p = 2yields p# = (np−p)/(n−p) = 4, cf. (1.37). This just agrees with the 4-power growth of theStefan-Boltzmann law at least in the sense that the traces |u|3u are in L1(Γ) if u ∈ W 1,2(Ω).Yet (2.55b) admits only (3−ε)-power growth of b(x, ·) which does not fit with the 4-power growthof Stefan-Boltzmann law.

51Hint: The coercivity assumption (2.124a) requires here sA(x, r)s ≥ ε1|s|p − k1, whichrequires, besides uniform positive definiteness of A, also p ≤ 2. Altogether, p = 2 is ultimatelyneeded. Note that p = 2 and (2.55a) need A(x, r) bounded, i.e. |aij(x, r)| ≤ C for any i, j =1, . . . , n. The condition (2.124b) holds trivially with k2 = 0.


particular, that p = 2 is needed and the qualification (2.57) of (g, h) means

g ∈⎧⎨⎩

L1(Ω),L1+ε(Ω),

L2n/(n+2)(Ω),h ∈

⎧⎨⎩L1(Γ) if n = 1,L1+ε(Γ) if n = 2,

L2−2/n(Γ) if n ≥ 3.(2.127)

In view of this, the pseudo-monotone approach has disadvantages:� direct usage of Leray-Lions Theorem 2.36 on the conventional Sobolev space

W 1,2(Ω) is limited to n ≤ 2 or to b2 ≡ 0,

� if n > 1, an artificial integrability of g and h is needed, contrary to thephysically natural requirement of a finite energy of heat sources, i.e. g∈L1(Ω),h∈L1(Γ),

� A must be bounded.

Modify the setting of Section 2.4.3 by replacing W 1,p(Ω) by

V ={v∈W 1,2(Ω); v|Γ∈L5(Γ)

}(2.128)

and show that V becomes a reflexive Banach space densely containing C∞(Ω) ifequipped with the norm ‖v‖ := ‖v‖W 1,2(Ω)+

∥∥v|Γ∥∥L5(Γ).52 Show that A : V → V ∗

defined by (2.126) is bounded and coercive. Make a limit passage though themonotone boundary term by Minty’s trick instead of the compactness.

Exercise 2.73 (Weak-continuity approach). Use V from (2.128) and Z = W 1,∞(Ω),assume

∃ε1 > 0 : sA(x, r)s ≥ ε1|s|2, (2.129a)

∃γ∈L2(Ω), ε > 0 : |A(x, r)| ≤ γ(x) + C|r|(2∗−ε)/2, (2.129b)

and show that A : V → Z∗ defined by (2.126) is weakly continuous; use the factthat L4(Γ) is an interpolant between L2(Γ) and L5(Γ).53

Exercise 2.74 (Galerkin method). Consider Vk a finite-dimensional subspace ofW 1,∞(Ω) nested for k → ∞ with a dense union in W 1,2(Ω) and traces dense in

52Hint: For n ≤ 2, simply V = W 1,2(Ω). For n ≥ 3, any Cauchy sequence {vk}k∈N in V has

a limit v in W 1,2(Ω) and {vk |Γ}k∈N converges in Lp# (Γ) to v|Γ, and simultaneously has somelimit w in L5(Γ) but necessarily v|Γ = w. As V is (isometrically isomorphic to) a closed subspacein a reflexive Banach space W 1,2(Ω) × L5(Γ), it is itself reflexive. Density of smooth functionscan be proved by standard mollifying procedure.

53Hint: Take uk such that uk ⇀ u in W 1,2(Ω) and uk|Γ ⇀ u|Γ in L5(Γ). Use W 1,2(Ω) �L2∗−ε(Ω) and then A(uk) → A(u) in L2(Ω;Rn×n) and ∇uk ⇀ ∇u weakly in L2(Ω;Rn), and,for v ∈ W 1,∞(Ω) =: Z, pass to the limit

∫Ω(∇uk)

A(x, uk)∇v dx→ ∫Ω(∇u)A(x, u)∇v dx. By

compactness of the trace operator, uk|Γ → u|Γ in Lp#−ε(Γ) ⊂ L2(Γ), realize that uk|Γ → u|Γin L4(Γ) because∥∥uk|Γ − u|Γ

∥∥L4(Γ)

≤ ∥∥uk|Γ − u|Γ∥∥5/6L5(Γ)

∥∥uk|Γ − u|Γ∥∥1/6

L2(Γ)→ 0.

Then |uk|3uk|Γ → |u|3u|Γ in L1(Γ), and∫Γ |uk|3ukv dS →

∫Γ |u|3uv dS for any v ∈ L∞(Γ).


L5(Γ), and thus in V from (2.128), too. Then the Galerkin method for (2.125) isdefined by:∫

Ω

(∇uk)A(x, uk)∇v − gv dx+

∫Γ

((b1 + b2|uk|3)uk − h

)v dS = 0 (2.130)

for all v∈Vk. Assume g ∈ L2∗′(Ω), h = b1θ+b2θ

4 with θ ∈ L5(Γ), see Example 2.71,and assuming existence of uk, show the a-priori estimate in V from (2.128) byputting v := uk into (2.130).54 Then, using the linearity of s �→ a(x, r, s) =A(x, r)s, make the limit passage directly in the Galerkin identity (2.130) by usingthe weak continuity as in Exercise 2.73.

Exercise 2.75 (Strong convergence). Assume again A bounded as in Exercise 2.72and, despite the lack of d-monotonicity of u �→ div

(A(x, u)∇u

), use strong mono-

tonicity of u �→ div(A(x, v)∇u

)for v fixed, and show uk → u in W 1,2(Ω); make

the limit passage in the boundary term by compactness55 if n ≤ 2, or treat it by

54Hint: By Holder’s and Young’s inequalities, this yields the estimate

ε1

∫Ω|∇uk|2dx+ b1

∫Γ|uk|2dS + b2

∫Γ|uk|5dS ≤

∫Ω(∇uk)

A(x, uk)∇ukdx

+

∫Γb1|uk|2 + b2|uk|5dS =

∫Ωgukdx+

∫Γ(b1θ + b2θ

4)ukdS

≤ ‖g‖Lp∗′

(Ω)‖uk‖Lp∗ (Ω) +

(‖b1‖L5/3(Γ)‖θ‖L5(Γ) + ‖b2‖L∞(Γ)‖ |θ|4‖5/4L5/4(Γ)

)‖uk‖L5(Γ)

≤ 1

4εN2‖g‖2

Lp∗′(Ω)

+ ε‖uk‖2W1,2(Ω)

+C(‖b1‖L5/3(Γ)‖θ‖L5(Γ) + ‖b2‖L∞(Γ)‖θ‖4L5(Γ)

)5/4+

1

2b2‖uk‖5L5(Γ)

where N is the norm of the embedding W 1,2(Ω) ⊂ L2∗ (Ω) and C is a sufficiently large constant,namely C = 29/(55b2), and ε < min(ε1, b1)/CP with CP the constant from the Poincare inequal-ity (1.56) with p = 2 = q. Then use (1.56) for the estimate of the left-hand side from below andabsorb the right-hand-side terms with uk in the left-hand side.

55Hint: Abbreviate b(u) = (b1+b2|u|3)u and ak := ∇(uk−u)A(uk)∇(uk−u), cf. (2.83). Then,use the Galerkin identity (2.130), i.e.

∫Ω∇(uk−vk)A(uk)∇ukdx =

∫Γ b(uk)(vk−uk) dS, to get∫

Ωak dx =

∫Ω∇(uk−u)

(A(uk)∇uk − A(u)∇u

) −∇(uk−u)(A(uk) − A(u)

)∇u dx

=

∫Ω∇(uk−vk)


)+∇(vk−u)

(A(uk)∇uk−A(u)∇u

)−∇(uk−u)

(A(uk)− A(u)

)∇udx =

∫Γ

(b(u)−b(uk)

)(uk−vk) dS

+

∫Ω∇(vk−u)


) −∇(uk−u)(A(uk)− A(u)

)∇u dx

for any vk ∈ Vk. In particular, take vk → u in W 1,2(Ω). For n ≤ 2, use compactness of the

trace operator W 1,2(Ω) → Lp#−ε(Γ) ⊂ L5(Γ) and push the first right-hand-side term to zero.Furthermore, use ∇vk → ∇u in L2(Ω;Rn) and A(uk)∇uk − A(u)∇u bounded in L2(Ω;Rn) topush the second term to zero. Finally, push the last expression to zero when using

(A(uk) −

A(u))∇u → 0 in L2(Ω;Rn) (note that one cannot rely on A(uk) → A(u) in L∞(Ω;Rn×n),


monotonicity if n = 3 when this term has the super-critical growth.56 Note that,for n = 3, the super-critical growth of the boundary term is such that, althoughbeing formally a lower-order term, it behaves like a highest-order term and mustbe treated by its monotonicity.57

Exercise 2.76 (Comparison principle). Put v := u− = min(u, 0) into the integralidentity (2.51) for the case of (2.125). Show that non-negativity of heat sources,i.e. h = b1θ + b2θ

4 ≥ 0 and g ≥ 0, implies the non-negativity of temperature,i.e. u ≥ 0.58 Assume g = 0 and 0 ≤ θ(·) ≤ θmax for a constant θmax > 0 and usev := (u − θmax)

+ in (2.51) to show that u(·) ≤ θmax almost everywhere.

Exercise 2.77 (Mixed boundary conditions). Perform the analysis by the Galerkinmethod of the mixed Dirichlet/Newton boundary-value problem59

however) and when assuming vk → u in W 1,2(Ω). Then ε1‖∇uk−∇u‖2L2(Ω;Rd)

≤ ∫Ωakdx → 0

and, as uk → u in L2(Ω) by Rellich-Kondrachov’s theorem 1.21, uk → u in W 1,2(Ω).56Hint: Use identity (2.130) and the previous notation of ak and b to write∫Ωak dx+

∫Γ

(b(uk)− b(u)

)(uk−u) dS

=

∫Ω∇(uk−vk)


)+∇(vk−u)


)− ∇(uk−u)

(A(uk)−A(u)

)∇u dx+

∫Γ

(b(uk)−b(u)

)(uk−vk) +

(b(uk)−b(u)

)(vk−u) dS

=

∫Γ−b(u)(uk−vk) + (b(uk)−b(u))(vk−u) dS +

∫Ω−∇(uk−vk)A(u)∇u

+∇(vk−u)(A(uk)∇uk − A(u)∇u

) −∇(uk−u)(A(uk)− A(u)

)∇u dx

= I(1)k + I

(2)k + I

(3)k + I

(4)k + I

(5)k .

Assume vk → u in W 1,2(Ω) and vk |Γ → u|Γ in L5(Γ). Use b(u) ∈ L5/4(Γ) and uk − vk ⇀ 0 in

L5(Γ) to show I(1)k → 0. Use {b(uk)}k∈N bounded in L5/4(Γ) and vk − u→ 0 in L5(Γ) to show

I(2)k :=

∫Γ(b(uk)− b(u))(vk −u) dS → 0. Push the remaining terms I

(3)k , I

(4)k , and I

(5)k as before.

Altogether, conclude uk → u in W 1,2(Ω). Moreover, conclude also ‖uk − u‖L5(Γ) → 0.57Hint: Realize the difficulties in pushing

∫Γ(b(u)− b(uk))(uk−vk) dS to zero if n ≥ 3 because

we have {uk}k∈N and {b(uk)}k∈N only bounded in L5(Γ) and L5/4(Γ), respectively, but nostrong convergence can be assumed in these spaces.

58Hint: Note that u− ∈ W 1,2(Ω) if u ∈ W 1,2(Ω) so v := u− is a legal test, cf. Proposition 1.28,and then

∫Ω(∇u)A(u)∇u− dx =

∫Ω(∇u−)A(u)∇u− dx due to (1.50). By this way, come to

the estimate

ε1

∫Ω|∇u−|2dx+ b1

∫Γ(u−)2dS ≤

∫Ω(∇u)A(u)∇u− dx

+

∫Γb1(u

−)2 + b2|u−|5dS =

∫Ωgu− dx+

∫Γhu− dS ≤ 0.

By the Poincare inequality (1.56), we get ‖u−‖W1,2(Ω) = 0, hence u− = 0 a.e. in Ω.59Hint: Instead of (2.128), use V =

{v∈W 1,2(Ω); v|ΓN ∈L5(ΓN), v|ΓD = 0

}, define Galerkin’s

approximate solution uk with approximate Dirichlet conditions uk|ΓD = ukD, and derive the a-

priori estimate by a test v := uk − wk where wk ∈ Vk, a finite-dimensional subspace of V , is

chosen so that wk|ΓN = ukD → uD in L2#(ΓD) and the sequence {wk}k∈N is bounded in V .


−div(A(x, u)∇u

)= g(x) in Ω,

νA(x, u)∇u = b1(x)(θ − u) + b2(x)(θ4 − |u|3u) on ΓN,

u|ΓD = uD on ΓD.

⎫⎪⎬⎪⎭ (2.131)

Exercise 2.78 (Heat-conductive plate). Perform the analysis by Galerkin’s methodof the problem

−div(A(x, u)∇u

)= c1(x)(θ − u) + c2(x)(θ

4 − |u|3u) in Ω,

u|Γ = uD on Γ.

}(2.132)

In the case n = 2, this problem has an interpretation of a plate conducting heat intangential direction with normal-direction temperature variations neglected, andbeing cooled/heated by convection and radiation and with fixed temperature onthe boundary as outlined on Figure 6b. Consider n ≤ 3, use the conventionalSobolev space W 1,2

0 (Ω), define Galerkin’s approximate solution uk with approxi-mate Dirichlet conditions uk|Γ = uk

Γ, and derive the a-priori estimate by a test byv := uk − wk with wk as in Exercise 2.77.

Example 2.79 (Special nonlinear media). Let us consider again the nonlinear heat-transfer problem (2.125) with A(x, r) = [aij(x, r)] in the special form

aij(x, r) = bij(x)κ(r) (2.133)

with B = [bij ] : Ω → Rn×n and κ : R → R+. Then the so-called Kirchhofftransformation employs the primitive function κ : R → R to κ, i.e. defined by

κ (r) :=

∫ r

0

κ(�)d� , (2.134)

and transforms the nonlinearity of (2.125) inside Ω to the (already nonlinear)boundary conditions. Indeed, B(x)∇ κ (u) = B(x)κ(u)∇u = A(x, u)∇u andB(x) ∂

∂ν κ (u) = B(x)κ(u) ∂∂νu = A(x, u) ∂

∂νu and, by a substitution w = κ (u),one transfers the nonlinearity from the equation on Ω to the boundary conditionswhich has been nonlinear even originally anyhow due to the Stefan-Boltzmannradiation term. Thus one gets the following semilinear equation for w:

−div(B(x)∇w

)= g in Ω,

B(x)∂w

∂ν+(b1 + b2| κ−1(w)|3

)κ−1(w) = h on Γ.

}(2.135)

We assume B : Ω → Rn×n measurable, bounded, and B(·) uniformly positivedefinite in the sense ξB(x)ξ ≥ β|ξ|2 for all ξ ∈ Rn and some β > 0. Further,we assume κ(·) ≥ ε > 0 measurable and bounded; note that this implies κ tobe continuous and increasing, and one-to-one with κ−1 Lipschitz continuous, inparticular having a linear growth. Furthermore, g and h satisfy (2.127). Again,ultimately p = 2, and one can show the coercivity. As the function R → R : r �→


b(x, r) :=(b1(x) + b2(x)| κ−1(r)|3) κ−1(r) is monotone for a.a. x ∈ Γ, we can use

the monotonicity technique. Then there is just one weak solution w ∈ W 1,2(Ω).By Proposition 1.28, u = κ−1(w) ∈ W 1,2(Ω) and this u solves the original prob-

lem in the weak sense. Moreover, (g, h) �→ u : Lp∗′(Ω) × Lp#′

(Γ) → W 1,2(Ω)is (norm×norm,norm)-continuous. Note that the heat-conductivity coefficient κneed not be assumed continuous.60

Example 2.80 (Heat transfer with advection). The heat equation in moving homo-geneous isotropic media, i.e. with advection by a prescribed velocity, say �v = �v(x),is

−div(κ(u)∇u

)+ c(u)�v · ∇u = g, (2.136)

where c is the heat capacity dependent on temperature. Let us consider, for simplic-ity, constant Dirichlet boundary conditions and use the Kirchhoff transformation(2.134), i.e. put w = κ (u) and using ∇ κ−1(w) = ∇w/κ( κ−1(w)), to arrive at

−Δw +C(w)�v · ∇w

K(w)= g in Ω,

w = 0 on Γ

⎫⎬⎭ (2.137)

where we abbreviated κ( κ−1(w)) =: K(w) and c( κ−1(w)) =: C(w); note thatwe can shift κ by a constant so that w = 0 can be considered on Γ. Note thatthe pointwise coercivity (2.92a) for p = 2 ≥ q > 1 is violated if c(x, r, s) =C(r)�v(x) · s/K(r). Assume the velocity field �v ∈ C1(Ω;Rn) as divergence free,which corresponds to a motion of an incompressible medium, cf. also the equations(6.26c), (12.44c), or (12.95b) below, one can consider an alternative setting withc(u)�v · ∇u = div(�v c (u)) = div(�v c (κ−1(w))) with c the primitive function of c.This leads to a(x, r, s) = s+ �v(x) c (κ−1(r)) which again need not satisfy (2.92a).

Exercise 2.81 (Uniqueness). Show uniqueness of the weak solution w to (2.135),and thus of u, as well. Try to show uniqueness in the general case (2.125) andrealize the difficulties if smallness of ‖u‖W 1,∞(Ω) is not guaranteed.

61

Exercise 2.82. Assume div�v ≤ 0 in Example 2.80 and show the coercivity ofthe respective A (in spite of this lack of any pointwise coercivity pointed outin Example 2.80) by derivation of an a-priori estimate again by a test by w.62

60A discontinuity of κ can indeed occur during various phase transformations, cf. [358] for adiscontinuity in the heat-conductivity coefficient κ within a recrystallization in steel.

61The uniqueness holds even for a general case (2.125) but the proof is rather technical, cf. [243].62Hint: For N the norm of the embedding W 1,2(Ω) ⊂ L2∗ (Ω), use Green’s Theorem 1.31 to

estimate∫Ω|∇w|2 dx ≤

∫Ω|∇w|2 − (div�v)

(∫ w(x)

0

C(ξ)

K(ξ)dξ

)dx =

∫Ω|∇w|2 + �v ·∇

(∫ w(x)

0

C(ξ)

K(ξ)dξ

)dx

=

∫Ω|∇w|2 +

(�v · ∇w)C(w)

K(w)dx =

∫Ωgw dx ≤ N‖w‖W1,2(Ω)‖g‖L2∗′

(Ω).


Furthermore, assuming Lipschitz continuity of κ, show uniqueness of a solution to(2.137) if �v is small enough in the L∞-norm.63

2.6.3 Quasilinear equations of type −div(|∇u|p−2∇u

)+c(u,∇u)=g

Here we will address quasilinear equations (2.45) with a(x, r, ·) or c(x, r, ·) nonlin-ear so that a limit passage in approximate solutions cannot be made by using mereweak convergence in ∇u and compactness in lower-order terms, unlike in semilin-ear equations scrutinized in Section 2.6.2. As a “training” quasilinear differentialoperator in the divergence form, we will frequently use

Δpu := div(|∇u|p−2∇u

)(2.138)

called the p-Laplacean; hence the usual Laplacean is what is called here 2-Laplacean. For p > 2 one gets a degenerate nonlinearity, while for p < 2a singular one, cf. Figure 9 on p.128 below. Note that, by using the formuladiv(vw) = v divw +∇v · w, (2.138) can equally be written in the form

div(|∇u|p−2∇u

)= |∇u|p−2Δu+ (p−2)|∇u|p−4(∇u)∇2u∇u. (2.139)

Example 2.83 (d-monotonicity of p-Laplacean). To be more specific, A = −Δp

will be understood here as a mapping W 1,p(Ω) → W 1,p(Ω)∗ corresponding to aNeumann-boundary-value problem, i.e.

〈A(u), v〉 =∫Ω

|∇u|p−2∇u · ∇v dx (2.140)

for any v ∈ W 1,p(Ω). For p > 1, the p-Laplacean is always d-monotone in the sense(2.1) with respect to the seminorm |u| := ‖∇u‖Lp(Ω;Rn), i.e.∫

Ω

(|∇u|p−2∇u−|∇v|p−2∇v) · (∇u−∇v) dx ≥ (d(|u|)−d(|v|))(|u|−|v|)

63Hint: Realizing that also [C/K](·) is Lipschitz continuous, with � denoting the Lipschitzconstant, we have∫

Ω�v ·

(C(w1)∇w1

K(w1)− C(w2)∇w2

K(w2)

)(w1 − w2) dx

=

∫Ω�v(C(w1)

K(w1)− C(w2)

K(w2)

)· ∇w1(w1−w2) dx+

∫Ω

C(w2)�v · ∇(w1−w2)

K(w2)(w1−w2) dx

≤ ‖�v‖L∞(Ω;Rn)

∥∥∥C(w1)

K(w1)− C(w2)

K(w2)

∥∥∥L4(Ω)

‖∇w1‖L2(Ω;Rn)‖w1 −w2‖L4(Ω)

+ ‖�v‖L∞(Ω;Rn)

∥∥∥C(w2)

K(w2)

∥∥∥L4(Ω)

‖∇w1 −∇w2‖L2(Ω;Rn)‖w1 − w2‖L4(Ω)

≤ ‖�v‖L∞(Ω;Rn)

(‖∇w1‖L2(Ω;Rn)�N

2 +Nmaxc(·)minκ(·)measn(Ω)1/4

)‖w1 −w2‖2W1,2(Ω)

,

where N is the norm of the embedding W 1,2(Ω) ⊂ L4(Ω) valid for n ≤ 3. For ‖�v‖L∞(Ω;Rn) smallenough, conclude that w1 = w2.


with d(ξ) = ξp−1, which can be proved simply by Holder’s inequality as follows:∫Ω

(|y|p−2y − |z|p−2z) · (y − z) dx

= ‖y‖pLp(Ω;Rn) −∫Ω

(|y|p−2y · z + |z|p−2z · y

)dx+ ‖z‖pLp(Ω;Rn)

≥ ‖y‖pLp(Ω;Rn) −∥∥|y|p−2y

∥∥Lp′(Ω;Rn)

‖z‖Lp(Ω;Rn)

− ∥∥|z|p−2z∥∥Lp′(Ω;Rn)

∥∥y∥∥Lp(Ω;Rn)

+∥∥z∥∥p

Lp(Ω;Rn)

=∥∥y∥∥p

Lp(Ω;Rn)− ∥∥y∥∥p−1

Lp(Ω;Rn)‖z‖Lp(Ω;Rn)

− ∥∥z∥∥p−1

Lp(Ω;Rn)

∥∥y∥∥Lp(Ω;Rn)

+∥∥z∥∥p

Lp(Ω;Rn)

=(∥∥y∥∥p−1

Lp(Ω;Rn)− ∥∥z∥∥p−1

Lp(Ω;Rn)

)(∥∥y∥∥Lp(Ω;Rn)

− ∥∥z∥∥Lp(Ω;Rn)

). (2.141)

For p ≥ 2, from the algebraic inequality64(|s|p−2s− |s|p−2s) · (s− s

) ≥ c(n, p)|s− s|p (2.142)

with some c(n, p) > 0, we obtain a uniform monotonicity on W 1,p0 (Ω) in the sense

(2.2) with ζ(z) = zp−1 (or with respect to the seminorm ‖∇·‖Lp(Ω;Rn) onW 1,p(Ω)):⟨A(u)−A(v), u−v

⟩=

∫Ω

(|∇u|p−2∇u−|∇v|p−2∇v)·∇(u−v) dx

≥ c(n, p)

∫Ω

|∇u−∇v|pdx.

It should be emphasized that, for p < 2, one has only 〈A(u)−A(v), u−v〉 ≥(p−1)

∫Ωmax(1+|∇u|, 1+|∇v|)p−2|∇u−∇v|2dx.65

Exercise 2.84 (Monotonicity of p-Laplacean). Realize that (2.138) corresponds toai(x, r, s) = |s|p−2si and verify the strict monotonicity (2.65) and (2.68a).66

Exercise 2.85 (Strong convergence in c(∇u)). Consider the Dirichlet boundary-value problem

−div(|∇u|p−2∇u

)+ c(x,∇u) = g in Ω,

u = 0 on Γ.

}(2.143)

For some ε > 0, assume the growth condition

∃γ∈L(p∗−ε)′(Ω) C∈R ∀(a.a.)x∈Ω ∀s∈Rn :∣∣c(x, s)∣∣ ≤ γ(x)+C|s|p−1−ε. (2.144)

64See DiBenedetto [120, Sect.I.4] or Hu and Papageorgiou [209, Part.I,Sect.3.1].65See Malek et al. [268, Sect.5.1.2].66Hint: Like (2.141), (|s|p−2s−|s|p−2s)·(s−s) = |s|p−|s|p−2s·s−|s|p−2s·s+|s|p ≥ |s|p −

|s|p−1|s| − |s|p−1|s|+ |s|p = (|s|p−1−|s|p−1)(|s|−|s|), hence (2.65) holds. If (|s|p−2s−|s|p−2s) ·(s−s) = 0, then |s| = |s|, and if s �= s, then |s|2 > s·s hence |s|p−|s|p−2s·s > 0, and similarly|s|p−|s|p−2s·s > 0, hence (|s|p−2s−|s|p−2s)·(s−s) > 0, a contradiction, proving (2.68a).


Formulate Galerkin’s approximation67 and prove the a-priori estimate in W 1,p0 (Ω)

by testing the Galerkin identity by v = uk68 and prove strong convergence of

{uk} in W 1,p0 (Ω) by using d-monotonicity of −Δp, following the scheme of Propo-

sition 2.20 with Remark 2.21 simplified by having boundedness guaranteed explic-itly through Lemma 2.31 instead of the Banach-Steinhaus principle through (2.36)and (2.42).69 Further, considering p = 2, formulate a Lipschitz-continuity condi-tion like (2.157) in Exercise 2.90 that would guarantee (uniform) monotonicity ofthe underlying mapping A.

Exercise 2.86 (Weak convergence in c(∇u)). Consider the boundary-value problem(2.143) in a more general form:

−div a(x,∇u) + c(x,∇u) = g in Ω,

u = 0 on Γ,

}(2.145)

with a(x, ·) : Rn → Rn strictly monotone. The Galerkin approximation looks as∫Ω

a(∇uk) · ∇v +(c(∇uk)− g

)v dx = 0 ∀v∈Vk. (2.146)

Assuming coercivity a(x, s) · s ≥ εa|s|p and the growth (2.144), prove the a-priori

67See (2.146) below for a(x, s) = |s|p−2s.68Hint: Use Holder’s inequality between Lp/(p−1−ε)(Ω) and Lq(Ω) with q = p/(1+ε) to esti-

mate

∥∥uk

∥∥pW

1,p0 (Ω)

=∥∥∇uk

∥∥pLp(Ω;Rn)

=

∫Ω

(g−c(∇uk)

)uk dx ≤

∫Ω

(|g|+γ+C|∇uk|p−1−ε)|uk| dx

≤ ∥∥|g|+ γ∥∥Lp∗′

(Ω)

∥∥uk

∥∥Lp∗(Ω)

+ C‖∇uk‖p−1−εLp(Ω)

‖uk‖Lq(Ω)

≤ Np∗∥∥|g|+ γ

∥∥Lp∗′

(Ω)

∥∥uk

∥∥W

1,p0 (Ω)

+ CNq

∥∥uk

∥∥p−ε

W1,p0 (Ω)

with Nq the norm of the embedding W 1,p(Ω) ⊂ Lq(Ω), and Np∗ with an analogous meaning.69Hint: Take a subsequence uk ⇀ u in W 1,p

0 (Ω). Use the norm ‖v‖W

1,p0 (Ω)

:= ‖∇v‖Lp(Ω;Rn)

and, by (2.141) and using still the abbreviation a(∇v) = |∇v|p−2∇v, estimate

(‖uk‖p−1

W1,p0 (Ω)

−‖v‖p−1

W1,p0 (Ω)

)(‖uk‖W1,p

0 (Ω)−‖v‖

W1,p0 (Ω)

)≤

∫Ω

(a(∇uk)−a(∇v)

) · ∇(uk−v) dx

=

∫Ωa(∇uk) · ∇(uk−vk) + a(∇uk) · ∇(vk−v) − a(∇v) · ∇(uk−v) dx

=

∫Ω

(g − c(∇uk)

)(uk−vk) + a(∇uk) · ∇(vk−v) − a(∇v) · ∇(uk−v) dx

with vk ∈ Vk. Assume vk → v. For v = u, uk − vk → u − u = 0 in Lp∗−ε(Ω) because of

the compact embedding W 1,p0 (Ω) � Lp∗−ε(Ω), and then

∫Ω c(∇uk)(uk − vk) dx → 0; note that

{c(∇uk)}k∈N is bounded in L(p∗−ε)′(Ω). Push the other terms to zero, too. Conclude that

uk → u in W 1,p0 (Ω). Then, having got the strong convergence ∇uk → ∇u, pass to the limit

directly in the Galerkin identity.


estimate by testing (2.146) by v = uk.70 Then prove weak convergence of the

Galerkin method as in (2.84).71

Exercise 2.87. Modify Exercises 2.85 and 2.86 for non-zero Dirichlet or Newtonboundary conditions.

Exercise 2.88 (Monotone case I). Consider the boundary-value problem (2.45)–(2.49) in the special case ai(x, r, s) := ai(x, s) and c(x, r, s) := c(x, r), i.e.

−div a(∇u) + c(u) = g on Ω,ν ·a(∇u) + b(u) = h on ΓN,

u|ΓD = uD on ΓD.

⎫⎬⎭ (2.147)

Assume that a(x, ·), b(x, ·), and c(x, ·) are monotone, coercive (say a(x, s)·s ≥ |s|p,b(x, 0) = 0, c(x, 0) = 0, and measn−1(ΓD) > 0) with basic growth conditions, i.e.(

a(x, s)− a(x, s))·(s−s) ≥ 0,

∃γa∈Lp′(Ω), Ca∈R : |a(x, s)| ≤ γa(x) + Ca|s|p−1, (2.148a)(

b(x, r) − b(x, r))(r − r) ≥ 0,

∃γb∈Lp#′(Γ), Cb∈R : |b(x, r)| ≤ γb(x) + Cb|r|p#−1, (2.148b)(

c(x, r) − c(x, r))(r − r) ≥ 0,

∃γc∈Lp∗′(Ω), Cc∈R : |c(x, r)| ≤ γc(x) + Cc|r|p∗−1, (2.148c)

and prove a-priori estimates72 and the convergence of Galerkin’s approximationsby Minty’s trick.73

70Hint: Estimate

εa‖uk‖pW

1,p0 (Ω)

= εa‖∇uk‖pLp(Ω;Rn)≤

∫Ωa(∇uk) · ∇ukdx ≤

∫Ω

(g − c(∇uk)

)ukdx

and finish it as in Exercise 2.85.71Hint: Prove limk→∞

∫Ω(a(∇uk)−a(∇u)) ·∇(uk −u) dx = 0 as in Exercise 2.85. Then, for a

selected subsequence, deduce c(∇uk)→ c(∇u) a.e. in Ω by the same way as done in (2.88), and

similarly also a(∇uk)→ a(∇u) a.e. in Ω. Then prove a(∇uk) ⇀ a(∇u) in Lp′(Ω) and c(∇uk) ⇀

c(∇u) in Lp∗′+ε(Ω) and pass to the limit directly in (2.146) for any v ∈ ⋃h>0 Vk without using

Minty’s trick. Finally, extend the resulted identity by continuity for any v ∈ W 1,p(Ω).72Hint: Denoting uD ∈ W 1,p(Ω) an extension of uD test the Galerkin identity determining

uk ∈ Vk by v := uk − uk where uk|Γ → uD in W 1,p(Ω)|Γ for k → ∞, {uk}k∈N bounded inW 1,p(Ω), uk ∈ Vk, Vk a finite-dimensional subspace of W 1,p(Ω). Arrive to∫

Ω|∇uk|pdx ≤

∫Ωa(∇uk) · ∇uk + c(uk)ukdx+

∫ΓN

b(uk)ukdS

=

∫Ωa(∇uk) · ∇uk + c(uk)uk + g(uk−uk) dx+

∫ΓN

b(uk)uk + h(uk−uk) dS

and then get uk estimated in W 1,p(Ω) by Holder’s inequality and Poincare’s inequality (1.57).Alternatively, use the a-priori shift as in Proposition 2.27.

73Hint: For v ∈ W 1,p(Ω), use vk → v in W 1,p(Ω), vk ∈ Vk, uk|ΓD = vk |ΓD , the monotonicity


Exercise 2.89 (Monotone case II). Consider A:W 1,max(2,p)(Ω) → W 1,max(2,p)(Ω)∗

given by⟨A(u), v

⟩=

∫Ω

(1 + |∇u|p−2)∇u · ∇v + c(u)v dx+

∫Γ

b(u)v dS (2.149)

so that the equation A(u) = f with f from (2.60) corresponds to the boundary-value problem for the regularized p-Laplacean:

−div((1 + |∇u|p−2)∇u

)+ c(x, u) = g in Ω,(

1 + |∇u|p−2)∂u∂ν

+ b(x, u) = h on Γ.

}(2.150)

Assume c(x, ·) strongly monotone and b(x, ·) either increasing or, if decreasing ata given point r, then being locally Lipschitz continuous with a constant −b :(

c(x, r) − c(x, r))(r − r) ≥ εc(r − r)2, (2.151)(

b(x, r) − b(x, r))(r − r) ≥ − −b (r − r)2. (2.152)

Show that A can be monotone even if b(x, ·) is not monotone; assume that74

−b ≤ N−2 min(1, εc). (2.153)

Show further strong monotonicity of A with respect to the W 1,2-norm if (2.153)holds as a strict inequality.

Exercise 2.90 (Monotone case III). Let A : W 1,max(2,p)(Ω) → W 1,max(2,p)(Ω)∗ begiven by⟨

A(u), v⟩=

∫Ω

(1 + |∇u|p−2

)∇u · ∇v + c(∇u)v dx+

∫Γ

b(u)v dS. (2.154)

and Galerkin’s identity

0 ≤∫Ω

(a(∇uk)− a(∇vk)

) · ∇(uk−vk) +(c(uk)− c(vk)

)(uk−vk) dx+

∫ΓN

(b(uk) − b(vk)

)×(uk−vk) dS =

∫Ω

(g − c(vk)

)(uk−vk)− a(∇vk) ·∇(uk−vk) dx+

∫ΓN

(h− b(vk)

)(uk−vk) dS

→∫Ω

(g − c(v)

)(u−v) − a(∇v) · ∇(u−v) dx+

∫ΓN

(h− b(v)

)(u−v) dS

and then put v := u± εw, divide it by ε > 0, and pass ε→ 0.74Hint: Indeed,⟨

A(u)− A(v), u− v⟩=

∫Ω|∇u−∇v|2 +

(|∇u|p−2∇u− |∇v|p−2∇v)·(∇u−∇v)

+(c(u)− c(v)

)(u− v) dx+

∫Γ

(b(u)− b(v)

)(u− v) dS

≥∫Ω|∇u−∇v|2 + εc(u− v)2dx−

∫Γ�−b (u− v)2dS

≥ min(1, εc)‖u−v‖2W1,2(Ω)

− �−b ‖u−v‖2L2(Γ)≥ (

min(1, εc) − �−b N2)‖u−v‖2

W1,2(Ω).


Note that the equation A(u) = f with f from (2.60) corresponds to the boundary-value problem

−div((1 + |∇u|p−2)∇u

)+ c(x,∇u) = g for x ∈ Ω,(

1 + |∇u|p−2)∂u∂ν

+ b(x, u) = h for x ∈ Γ.

⎫⎬⎭ (2.155)

Assume b(x, ·) strongly monotone and c(x, ·) Lipschitz continuous, i.e.(b(x, r) − b(x, r)

)(r − r) ≥ εb|r − r|2, (2.156)∣∣c(x, s)− c(x, s)∣∣ ≤ c|s− s|, (2.157)

and show monotonicity of A if c is sufficiently small, despite that u �→ c(∇u)alone would not allow for any monotone structure.75 In particular, if c is smallenough, realize that A is strictly monotone and uniqueness of the solution follows.

Exercise 2.91 (Monotone case IV: advection). Consider a special case of (2.155)with c(x, s) := �v(x) · s with �v : Ω → Rn being a prescribed velocity field. Assumediv�v ≤ 0 (as in Exercise 2.82) and �v|Γ · ν ≥ 0, and show that A enjoys themonotonicity76 even if there is no point-wise monotonicity.

Exercise 2.92. Consider the following boundary-value problem:

−div(|∇u|p−2∇u+ a0(x, u)

)= g in Ω,

|∇u|p−2 ∂u

∂ν+ b0(x, u) = h on Γ.

⎫⎪⎬⎪⎭ (2.158)

75Hint: Estimate⟨A(u)− A(v), u− v

⟩=

∫Ω

((1 + |∇u|p−2)∇u− (1 + |∇v|p−2)∇v

)·∇(u− v)

+(c(∇u)− c(∇v)

)(u− v) dx+

∫Γ

(b(u) − b(v)

)(u− v) dS

≥ ‖∇u−∇v‖2L2(Ω;Rn)

− ‖c(∇u)− c(∇v)‖L2(Ω)‖u− v‖L2(Ω) + εb‖u− v‖2L2(Γ)

≥ ‖∇u−∇v‖2L2(Ω;Rn)

− �c‖∇u−∇v‖L2(Ω;Rn)‖u−v‖L2(Ω) + εb‖u− v‖2L2(Γ)

≥(1− �2c

2δ

)‖∇u−∇v‖2

L2(Ω;Rn)+ εb‖u−v‖2L2(Γ)

− δ

2‖u−v‖2

L2(Ω)

≥ C−1P

min

(1− �2c

2δ, εb

)‖u−v‖2

W1,2(Ω)− δ

2N2‖u−v‖2

W1,2(Ω),

with N the norm of the embedding W 1,2(Ω) ⊂ L2(Ω) and CP the constant from the Poincareinequality (1.56) with p = 2 = q and ΓN = Γ. If �c is so small that there is some δ > 0 such that

min

(1− �2c

2δ, εb

)≥ δ

2N2CP ,

the monotonicity of A follows.76Hint: By using Green’s formula, the monotonicity of this linear term is based on the estimate:∫

Ω(�v · ∇u)u dx =

1

2

∫Ω�v · ∇u2dx =

1

2

∫Γ

(�v · ν)u2dS − 1

2

∫Ω

(div�v

)u2dx ≥ 0.


Assume the basic growth condition: |a0(x, r)| ≤ γ(x) + C|r|p∗/p′for some

γ ∈ Lp′(Ω) and formulate a definition of the weak solution; denote: b(x, r) :=

b0(x, r)−a0(x, r)·ν(x). Prove that u �→div(a0(x, u)) : W1,p(Ω) → W 1,p(Ω)∗ is a to-

tally continuous mapping (which allows us to use Theorem 2.6 with Corollary 2.12to get the existence of a weak solution). Further prove the a-priori estimate by test-ing by u.77 Prove convergence of the Galerkin approximation via Minty’s trick,and alternatively strong convergence and direct limit passage without Minty’strick. Show uniqueness of the weak solution for Lipschitz continuous a0(x, ·) witha small Lipschitz constant. Make the modification for the Dirichlet boundarycondition.78

Example 2.93 (Banach fixed-point technique). Consider the boundary-value prob-lem (2.147)) and assume the strong monotonicity of a(x, ·) and, e.g., of c(x, ·) butno monotonicity of b(x, ·), i.e.

(a(x, s)−a(x, s)

) · (s−s) ≥ εa|s− s|2, (2.159a)(c(x, r) − c(x, r)

)(r − r) ≥ εc(r − r)2, (2.159b)

and the Lipschitz continuity

|a(x, s) − a(x, s)| ≤ a|s− s|, (2.160a)

− −b (r − r)2 ≤ (b(x, r) − b(x, r))(r − r) ≤ +b (r − r)2, (2.160b)

|c(x, r) − c(x, r)| ≤ c|r − r|, (2.160c)

with some +b ≥ −b ≥ 0; note that b(x, ·) is Lipschitz continuous with the constant +b . Then one can use the Banach fixed-point Theorem 1.12 technique based onthe contractiveness of the mapping Tε from (2.43) where the Lipschitz constant of A can be estimated as:79

77Hint: Realize that∫Ω|∇u|pdx+

∫Γb0(u)u dS =

∫Ω−a0(u) · ∇u+ gudx+

∫Γ

(a0(u) · ν + h

)u dS.

Assume b0(x, r)r ≥ |r|p, and estimate u by assuming further |a0(x, r)| ≤ γ(x) + C|r|p−1−ε.78Hint: Denoting a (x, r) =

(a 1(x, r), . . . , an(x, r)

)the component-wise primitive functions

to a0(x, r) =(a1(x, r), . . . , an(x, r)

)and realizing that now u|Γ = uD, by Green’s Theorem 1.31,

one gets

∫Ωa0(x, u)·∇udx =

∫Ω

n∑i=1

∂

∂xia i(x, u) dx =

∫Γ

n∑i=1

a i(x, u)νi(x) dS =

∫Γa (uD)·ν dS = const.

79Cf. also (4.17) below.


‖A(u)−A(v)‖W 1,2(Ω)∗ = sup‖z‖W1,2(Ω)≤1

〈A(u)−A(v), z〉

= sup‖z‖W1,2(Ω)≤1

∫Ω

(a(∇u)−a(∇v)

)·∇z +(c(u)−c(v)

)zdx+

∫ΓN

(b(u)−b(v)

)zdS

≤ sup‖z‖W1,2(Ω)≤1

( a‖∇u−∇v‖L2(Ω;Rn)‖∇z‖L2(Ω;Rn)

+ c‖u− v‖L2(Ω)‖z‖L2(Ω) + +b ‖u− v‖L2(ΓN)‖z‖L2(ΓN)

)≤ (√2max( a, c) +N2 +b

)‖u− v‖W 1,2(Ω) =: ‖u− v‖W 1,2(Ω)

while the constant δ in the strong monotonicity of A can be estimated as〈A(u) − A(v), u − v〉 ≥ (

min(εc, εa) − N2 −b)‖u − v‖2W 1,2(Ω) =: δ‖u − v‖2W 1,2(Ω);

cf. Exercise 2.89. Then, by Proposition 2.22, Tε from (2.43) with J : W 1,2(Ω) →W 1,2(Ω)∗ defined by80

〈J(u), v〉 =∫Ω

∇u·∇v + uv dx (2.161)

is a contraction provided ε > 0 satisfies81

ε < 2min(εc, εa)−N2 −b(√2max( a, c) +N2 +b

)2 . (2.162)

Exercise 2.94. Modify the above Example 2.93 for Dirichlet boundary conditions82

and/or the term c(∇u) instead of c(u)83.

Example 2.95 (Limit passage in coefficients). Consider the problem from Exam-ple 2.93 modified, for simplicity, as in Exercise 2.94 with zero Dirichlet boundaryconditions. Assume s �→ a(x, s) and r �→ c(x, r) monotone, a(x, s) · s+ c(x, r) · r ≥ε0|s|p − C, |a(x, s)| ≤ γ(x) + C|s|p−1 with γ ∈ Lp′

(Ω) and 1 < p ≤ 2. Such aproblem does not satisfy (2.159) and (2.160a,c). Therefore, we approximate a andc respectively by some aε and cε which will satisfy both (2.159) and (2.160a,c) andsuch that aε(x, ·) → a(x, ·) uniformly on bounded sets in Rn, and cε(x, ·) → c(x, ·)uniformly on bounded sets in R, and such that the collection {(aε, cε)}ε>0 is uni-formly coercive in the sense

∃δ > 0 ∀ε > 0 : aε(x, s) · s+ cε(x, r) · r ≥ δ|s|p − 1/δ. (2.163)

80Note that 〈J(u), u〉 = ‖u‖2W1,2(Ω)

and also ‖u‖W1,2(Ω) = ‖J(u)‖W1,2(Ω)∗ if one considers

the standard norm ‖u‖W1,2(Ω) =√‖∇u‖2

L2(Ω;Rn)+ ‖u‖2

L2(Ω); cf. Remark 3.15.

81Cf. (2.44) on p. 42.82Hint: Instead of (2.161) use 〈J(u), v〉 = ∫

Ω∇u ·∇v dx, cf. Proposition 3.14.83Hint: In case of Newton boundary conditions, b(x, ·) has to be strongly monotone as in

Exercise 2.90.


E.g. one can put aε(x, ·) := Y Mn,ε(a(x, ·)) and cε(x, ·) := Y M

1,ε(c(x, ·)) where Y Mn,ε :

Rn → Rn denotes a suitablemodification of Yosida’s approximation Yn,ε:Rn → Rn

defined by

[Y M

n,ε(f)](s) :=

[Yn,ε

(f +

ε2

1−εIn

)](s) with (2.164a)

[Yn,ε(f)

](s) :=

s− (In+εf)−1

(s)

ε(2.164b)

and In the identity on Rn; cf. also Remark 5.18 below. Unlike the mere Yosidaapproximation Yn,ε, the regularization (2.164) turns monotonicity to strong mono-tonicity; note also that Y M

n,ε(In) = In.

s r

00 11

11

−1−0.5

a(s) = |s|1/2sc(r) = r3

aε

cε, ε = 0.3ε = 0.1

ε = 0.1ε = 0.3

Figure 7. A regularization of the nonlinearities a(x, s) = |s|1/2s and c(x, r) = r3

that makes them both strongly monotone and Lipschitz continuous.

Then we can obtain the weak solution uε ∈ W 1,2(Ω) of the approximate problem

−div aε(∇u) + cε(u) = g in Ω,

u = 0 on Γ

}(2.165)

constructively by Example 2.93 (modified as in Exercise 2.94). The convergenceof uε ∈ W 1,2

0 (Ω) for ε → 0 relies on an a-priori estimate in W 1,p0 (Ω) which is

uniform with respect to ε > 0 due to (2.163), and then a selection of a subsequenceuε ⇀ u in W 1,p(Ω). Note that, as p ≤ 2, we have W 1,p

0 (Ω) ⊃ W 1,20 (Ω). Taking

v ∈ W 1,∞0 (Ω) and using monotonicity, we obtain

0 ≤∫Ω

(aε(∇uε)−aε(∇v)

)·(∇uε−∇v) +(cε(uε)−cε(v)

)(uε−v) dx

=

∫Ω

(g − cε(v)

)(uε − v)− aε(∇v)·(∇uε −∇v) dx

→∫Ω

(g − c(v)

)(u− v)− a(∇v)·(∇u−∇v) dx (2.166)

for ε → 0, where we used aε(∇v) → a(∇v) in L∞(Ω;Rn). Then we can pass vto v ∈ W 1,p

0 (Ω); by density of W 1,∞0 (Ω) in W 1,p

0 (Ω), cf. Theorem 1.25, v can be


considered arbitrary. By continuity of the Nemytskiı mappings Na : Lp(Ω;Rn) →Lp′

(Ω;Rn) and Nc : Lp∗(Ω) → Lp∗′

(Ω), from (2.166) we get∫Ω(g − c(v))(u − v)−

a(∇v) · (∇u−∇v) dx ≥ 0. Eventually, by Minty’s trick, we conclude that u solves(2.165); cf. Lemma 2.13.

Remark 2.96 (Constructivity). Let us still point out that, by combining the Ba-nach fixed-point iterations as in Example 2.93 with some coefficient approxima-tion as in Example 2.95, one can solve problems as (2.147) under quite weakassumptions rather constructively, without any Brouwer’s fixed-point argument,cf. Remark 2.7. In case of strict monotonicity in (2.147), the whole sequence ofapproximate solutions converges.

Exercise 2.97. Modify Example 2.95 for the case of Newton boundary conditions.

Exercise 2.98. Add a term div(b(x, u,∇u,∇2u)

)here with b : Ω×R×Rn×Rn×n →

Rn into (2.97) and modify (2.99) and Propositions 2.42 and 2.43.

Exercise 2.99. Realizing that only four out of all six combinations of derivativesup to 3rd-order on the boundary have been used in (2.100), (2.101), (2.105), and(2.107), identify the remaining two combinations and explain why they are notcompatible with a consistent and selective weak formulation.84

Exercise 2.100 (Singular higher-order perturbations). Consider the weak solutionuε ∈ W 2,2(Ω) ∩W 1,p(Ω) of the problem

div(εdiv∇2u− |∇u|p−2∇u

)= g in Ω,

∂u∂ν = u = 0 on Γ.

}(2.167)

Prove the a-priori estimates∥∥uε

∥∥W 1,p(Ω)

≤ C,∥∥uε

∥∥W 2,2(Ω)

≤ C/√ε. (2.168)

By using Minty’s trick based on the monotonicity of the mapping εdiv2∇2 −Δp,

prove the weak convergence uε ⇀ u in W 1,p0 (Ω) to the solution of the boundary-

value problem div(|∇u|p−2∇u) + g = 0 and u = 0 on Γ.85 Alternatively, make

84Hint: These two wrong options would exactly over-determine either the first or the secondboundary term in (2.104).

85Hint: Taking into account the identity∫Ω |∇uε|p−2∇uε·∇v + ε∇2uε:∇2v − gvdx = 0, use

‖ε∇2uε‖L2(Ω;Rn×n) = O(√ε) and, for any v ∈W 2,2

0 (Ω) ∩W 1,p(Ω), show

0 ≤∫Ω

(|∇uε|p−2∇uε − |∇v|p−2∇v)·∇(uε − v) + ε

∣∣∇2uε −∇2v∣∣2 dx

=

∫Ωg(uε−v)−|∇v|p−2∇v·∇(uε−v)−ε∇2v:∇2(uε−v) dx→

∫Ωg(u−v)−|∇v|p−2∇v·∇(u−v) dx.

Then extend the limit identity by continuity for all v ∈ W 1,p0 (Ω), and use v := u ± εz and

accomplish it by Minty’s trick.

2.7. Excursion to regularity for semilinear equations 85

it by Minty’s trick based on the monotonicity of the mapping −Δp.86 Show also

the strong convergence uε → u in W 1,p0 (Ω) by using d-monotonicity of −Δp.

87

Modify it by considering also term c(∇u) as in Example 2.85 and/or Newton-typeboundary conditions, or a quasilinear regularizing term as in Example 2.46.

2.7 Excursion to regularity for semilinear equations

By regularity we understand, in general, that the weak solution has some addi-tional differentiability properties as a consequence of some additional qualificationof data, i.e. in case of the boundary-value problem (2.45)–(2.49) a certain differen-tiability of a, b, c, g, and h, and a qualification of Ω as smoothness or restrictionson angles of possible corners. This represents usually a difficult task and there areexamples showing that, in case of higher-order equations or systems of equations,any smoothness of the data need not imply an additional smoothness of weak so-lutions. Regularity theory is a broad and still developing area which determines alot of investigations in particular in systems of nonlinear equations and in numer-ical analysis, and the exposition presented below is to be understood as only anabsolutely minimal excursion into this area.

We will confine ourselves to W k,2-type regularity for semilinear equationsand we start with a so-called interior regularity88 for the linear equation

86Hint: Again first for any v ∈W 2,20 (Ω) ∩W 1,p(Ω), calculate

0 ≤∫Ω

(|∇uε|p−2∇uε − |∇v|p−2∇v)·∇(uε − v) dx

=

∫Ωg(uε−v) − |∇v|p−2∇v·∇(uε−v) − ε∇2uε:∇2(uε−v) dx

≤∫Ωg(uε−v)−|∇v|p−2∇v·∇(uε−v)+ε∇2uε:∇2v dx→

∫Ωg(u−v)−|∇v|p−2∇v·∇(u−v) dx.

87Hint: Using Example 2.83, for any v ∈ W 2,20 (Ω) ∩W 1,p(Ω), show(∥∥∇uε

∥∥p−1

Lp(Ω;Rn)− ∥∥∇u

∥∥p−1

Lp(Ω;Rn)

)(∥∥∇uε

∥∥Lp(Ω;Rn)

− ∥∥∇u∥∥Lp(Ω;Rn)

)≤

∫Ω

(|∇uε|p−2∇uε − |∇u|p−2∇u)·∇(uε − u) dx

=

∫Ωg(uε−v) − ε∇2uε:∇2(uε−v) − |∇u|p−2∇u·∇(uε−u) + |∇uε|p−2∇uε·∇(v−u) dx

≤∫Ωg(uε−v) + ε∇2uε:∇2v − |∇u|p−2∇u·∇(uε−u) + |∇uε|p−2∇uε·∇(v−u) dx

→∫Ωg(u−v) + ξ·∇(v−u) dx

with some ξ ∈ Lp′(Ω;Rn) being a weak limit of (a subsequence) of |∇uε|p−2∇uε. Pushing v → uin W 1,p(Ω) makes the last expression arbitrarily close to zero, which shows ‖∇uε‖Lp(Ω;Rn) →‖∇u‖Lp(Ω;Rn), hence the strong convergence uε → u.

88This means we get estimates only in subdomains of Ω having a positive distance from Γ.


n∑i,j=1

∂

∂xi

(aij(x)

∂u

∂xj

)= g(x) on Ω (2.169)

with nonspecified boundary conditions. By a weak solution to (2.169) we willnaturally understand u ∈ W 1,2(Ω) such that

∫Ω(∇u)A∇v − gv dx = 0 for all

v ∈ W 1,20 (Ω) where A : Ω �→ Rn×n : x �→ A(x) = [aij(x)]

ni,j=1.

Proposition 2.101 (Interior W 2,2-regularity). Let A ∈ C1(Ω;Rn×n) satisfy

∃δ > 0 ∀ζ∈Rn ∀(a.a.)x∈Ω : ζA(x) ζ ≥ δ|ζ|2, (2.170)

g ∈ L2loc(Ω), and let u be a weak solution to (2.169). Then u ∈ W 2,2

loc (Ω). Moreover,for any open sets O,O2 ⊂ Rn satisfying O ⊂ O2 and O2 ⊂ Ω, it holds that

‖u‖W 2,2(O) ≤ C(‖g‖L2(O2) + ‖u‖L2(Ω)

)(2.171)

with C = C(O,O2, ‖A‖C1(Ω;Rn×n)

).

As the rigorous proof is very technical and not easy to observe, we begin witha heuristic one. Take still an open set O1 such that O ⊂ O1 and O1 ⊂ O2, and asmooth “cut-off function” ζ : Ω → [0, 1] such that χO ≤ ζ ≤ χO1 . Then, for a testfunction

v :=∂

∂xk

(ζ2

∂u

∂xk

)(2.172)

with k = 1, . . . , n, by using Green’s Theorem 1.30, we have formally the identity∫O1

n∑i,j=1

aij∂u

∂xj

∂

∂xi

(∂

∂xk

(ζ2

∂u

∂xk

))dx

= −∫O1

n∑i,j=1

∂

∂xk

(aij

∂u

∂xj

) ∂

∂xi

(ζ2

∂u

∂xk

)dx

= −∫O1

n∑i,j=1

(∂aij∂xk

∂u

∂xj+ aij

∂2u

∂xj∂xk

)(ζ2

∂2u

∂xi∂xk+ 2ζ

∂ζ

∂xi

∂u

∂xk

)dx. (2.173)

The identity (2.173) leads to the estimate

δ∥∥∥ζ∇ ∂u

∂xk

∥∥∥2L2(O1;Rn)

= δ

∫O1

n∑i=1

ζ2∣∣∣ ∂2u

∂xi∂xk

∣∣∣2dx≤∫O1

n∑i,j=1

ζ2aij∂2u

∂xi∂xk

∂2u

∂xj∂xkdx = −

∫O1

n∑i,j=1

ζ2∂aij∂xk

∂u

∂xj

∂2u

∂xi∂xk

+2ζ∂ζ

∂xi

∂u

∂xk

(∂aij∂xk

∂u

∂xj+ aij

∂2u

∂xj∂xk

)+ g

(∂

∂xk

(ζ2

∂u

∂xk

))dx


≤∫O1

n∑i,j=1

‖aij‖C1(Ω)

(ζ2∣∣∣ ∂u∂xj

∣∣∣ ∣∣∣ ∂2u

∂xi∂xk

∣∣∣+2ζ‖ζ‖C1(Ω)

∣∣∣ ∂u∂xk

∣∣∣(∣∣∣ ∂u∂xj

∣∣∣+ ∣∣∣ ∂2u

∂xj∂xk

∣∣∣))dx+ ‖g‖L2(O1)

∥∥∥2ζ∇ζ∂u

∂xk+ ζ2∇ ∂u

∂xk

∥∥∥L2(O1;Rn)

≤ C1

(‖∇u‖L2(O1;Rn) + ‖g‖L2(O1)

)(∥∥∥ζ∇ ∂u

∂xk

∥∥∥L2(O1;Rn)

+∥∥∥ζ ∂u

∂xk

∥∥∥L2(O1)

)≤ δ

2

∥∥∥ζ∇ ∂u

∂xk

∥∥∥2L2(O1;Rn)

+(C2

1

δ+

3

2

)(‖∇u‖2L2(O1;Rn) + ‖g‖2L2(O1)

)(2.174)

with C1 depending on ‖[aij ]ni,j=1‖C1(Ω;Rn×n) and ‖ζ‖C1(Ω). Then, letting k rangeover 1, .., n, we obtain

‖u‖W 2,2(O) ≤ C2

(‖g‖L2(O1) + ‖u‖W 1,2(O1)

). (2.175)

Finally, using a smooth “cut-off function” η : Ω → [0, 1] such that χO1 ≤ η ≤χO2 and the test-function v = ηu, we get δ‖∇u‖2L2(O1;Rn) ≤ δ

∫Ω η|∇u|2dx ≤∫

Ω ηgu dx ≤ 12‖g‖2L2(O2)

+ 12‖u‖2L2(Ω), which eventually leads to (2.171). The rigor-

ous proof is, however, more complicated because (2.172) is not a legal test functionunless we know that u ∈ W 2,2

loc (Ω), which is just what we want to prove.

Sketch of the proof of Proposition 2.101. We introduce the difference operator Dεk

defined by[Dε

ku](x) :=

u(x+ εek)− u(x)

ε, ε �= 0, [ek]i :=

{1 if i = k,0 if i �= k,

(2.176)

and use the test functionv := D−ε

k

(ζ2Dε

ku)

(2.177)

with k = 1, . . . , n. Note that, contrary to (2.172), now v ∈ W 1,20 (Ω) is a legal test

function. The analog of Green’s Theorem 1.30 is now∫Ω

vD−εk w dx =

∫Ω

v(x)w(x − εek)− w(x)

εdx

=1

ε

∫Ω

v(x)w(x − εek) dx− 1

ε

∫Ω

v(x)w(x) dx

=1

ε

∫Ω

v(x + εek)w(x) dx − 1

ε

∫Ω

v(x)w(x) dx = −∫Ω

wDεkv dx (2.178)

if |ε| is smaller than the distance ε0 of Γ from O1; note that v vanishes on Ω \O1.Moreover, by simple algebra, we have the formula

Dεk(vw) = SεkvD

εkw + wDε

kv (2.179)


with the “shift” operator Sεk defined by[Sεkv

](x) := v(x + εek). The analog of

(2.173) now reads as∫O1

n∑i,j=1

aij∂u

∂xj

∂

∂xi

(D−ε

k

(ζ2Dε

ku))

dx

= −∫O1

n∑i,j=1

Dεk

(aij

∂u

∂xj

) ∂

∂xi

(ζ2Dε

ku)dx

= −∫O1

n∑i,j=1

(Dε

kaij∂u

∂xj+ SεkaijD

εk

∂u

∂xj

)(ζ2Dε

k

∂u

∂xi+ 2ζ

∂ζ

∂xiDε

ku)dx.

(2.180)

We also use that ‖Dεkv‖L2(Ω1) ≤ ‖∇v‖L2(Ω) if |ε| ≤ ε0 := dist(O1,Γ).

89 Then theanalog of (2.174) reads as

δ∥∥ζ Dε

k∇u∥∥2L2(O1;Rn)

≤ δ

2

∥∥ζ Dεk∇u

∥∥2L2(O1;Rn)

+(C2

1

δ+

3

2

)(∥∥∇u∥∥2L2(O1;Rn)

+ ‖g‖2L2(O1)

). (2.181)

Hence the sequence (selected from) {ζ Dεk∇u}0<ε≤ε0 is bounded in L2(O1;Rn)

and converges, possibly as a subsequence, weakly to some w in L2(O1;Rn). In thesense of distributions, it must hold that w = ζ ∂

∂xk∇u.90 In particular, ∂

∂xk∇u ∈

L2(O;Rn) and, if considering k = 1, . . . , n, we have obtained (2.175). Then (2.171)follows as outlined in the heuristics. �

Proposition 2.102 (Interior W 3,2-regularity). Let A ∈ C1(Ω;Rn×n) ∩W 2,q(Ω;Rn×n) with q = 2∗2/(2∗ − 2) with 2∗ from (1.34) satisfy (2.170), andlet g ∈ W 1,2

loc (Ω), and let u be a weak solution to (2.169). Then u ∈ W 3,2loc (Ω).

Moreover, for any open sets O,O2 ⊂ Rn satisfying O ⊂ O2 and O2 ⊂ Ω, it holdsthat

‖u‖W 3,2(O) ≤ C(‖g‖W 1,2(O2) + ‖u‖L2(Ω)

)(2.182)

with C = C(O, ‖A‖C1(Ω;Rn×n)∩W 2,q(Ω;Rn×n)

).

Proof. Applying ∂∂xk

to (2.169), we obtain

n∑i,j=1

∂

∂xi

(aij

∂2u

∂xj∂xk

)=

∂g

∂xk−

n∑i,j=1

( ∂2aij∂xi∂xk

∂u

∂xj+

∂aij∂xk

∂2u

∂xj∂xi

)(2.183)

89It holds that [Dεkv](x) =

∫ 10

∂∂xk

(v + τεek)dτ so that, by Holder inequality, we obtain

‖Dεkv‖2L2(Ω1)

=∫Ω1

∣∣ ∫ 10

∂∂xk

(v + τεek)dτ∣∣2dx ≤ ∫

Ω

∣∣∇v∣∣2dx.

90For any v ∈ D(O) it holds that limε→0∫Ω1

(ζDεk∇u)v dx = limε→0−

∫ΩD−ε

k (ζv)∇u dx =

− ∫Ω

∂∂xk

(ζv)∇u dx = − ∫O

∂∂xk

v∇u dx.


in Ω. Note that, by Proposition 2.101, u ∈ W 2,2loc (Ω) and therefore (2.183) has

indeed a good “weak” sense: z := ∂∂xk

u is a weak solution to (2.169) with z

instead of u and with ∂∂xk

g − div(( ∂∂xk

A)∇u) − ( ∂∂xk

A)∇2u ∈ L2loc(Ω) instead of

g. Hence ∂∂xk

u ∈ W 2,2loc (Ω). �

For linear equations as (2.169) the process suggested in (2.183) can be

iterated for k = 4, . . . to obtain W k,2loc -regularity under the assumption that

A ∈ Ck−2(Ω;Rn×n) ∩ W k−1,q(Ω;Rn×n) and g ∈ W k−2,2(Ω). This differs fromnonlinear equations where the regularity has usually a natural bound. Here, weconfine ourselves to semilinear equations where results for linear equations candirectly be exploited. To be more specific, we will handle the equation

n∑i,j=1

∂

∂xi

(aij(x)

∂u

∂xj+ a0i(u)

)+ c0(∇u) + |u|q−2u = g(x) on Ω (2.184)

again with unspecified boundary conditions. By a weak solution to (2.184) we willnaturally understand u ∈ W 1,2(Ω) such that

∫Ω

((∇u)A+a0(u)

)·∇v+(c0(∇u)+

|u|q−2u− g)v dx = 0 for all v ∈ W 1,2

0 (Ω).

Proposition 2.103 (Regularity for semilinear equations).

(i) Let A ∈ C1(Ω;Rn×n) satisfy (2.170), let 1 < q ≤ (2n− 2)/(n− 2) for n ≥ 3(or q > 1 arbitrary if n ≤ 2), a0 : R → Rn be Lipschitz continuous, c0 haveat most linear growth, and g∈L2

loc(Ω). Then any weak solution u∈W 1,2(Ω)

to (2.184) satisfies also u∈W 2,2loc (Ω).

(ii) Moreover, let, in addition, A ∈W 2,max(2,n+ε)(Ω;Rn×n) with ε > 0 if n = 2(otherwise ε = 0 is allowed), and let also q ≥ 2, a0∈C2(R;Rn) with

a′′0 : R → Rn

⎧⎨⎩having arbitrary growth if n ≤ 3,being bounded if n = 4,≡ 0 if n ≥ 5,

(2.185)

c0 : Rn → R be Lipschitz continuous, and g∈W 1,2loc (Ω). Then any weak solution

u∈W 1,2(Ω) to (2.184) belongs also to W 3,2loc (Ω).

Proof. Note that u ∈ W 1,2(Ω) implies div(a0(u)) = a′0(u)∇u =∑n

i=1 a′0i(u)

∂∂xi

u

∈ L2(Ω) if a0 ∈ W 1,∞(R;Rn) as assumed, cf. Proposition 1.28. Also, c0(∇u) ∈L2(Ω) because of the linear growth of c0, and eventually |u|q−2u ∈ L2∗/(q−1)(Ω) ⊂L2(Ω) if 1 < q ≤ (2n− 2)/(n− 2) (or q > 1 arbitrary if n ≤ 2). Noting also thatthe exponent 2∗2/(2∗− 2) equals max(2, n) if n �= 2, or is greater than 2 if n = 2,we can use simply Proposition 2.101 with g being now g1 := g − div(a0(u)) −c0(∇u)− |u|q−2u ∈ L2(Ω). The point (i) is thus proved.


Assuming the additional data qualification as specified in the point (ii), wewant to show that g1 ∈ W 1,2

loc (Ω). For i = 1, . . . , n, we have

∂g1∂xi

=∂g

∂xi−

n∑j=1

(a′0j(u)

∂2u

∂xi∂xj

+ a′′0j(u)∂u

∂xi

∂u

∂xj+

∂c0∂si

(∇u)∂2u

∂xi∂xj

)− (q − 1)|u|q−2 ∂u

∂xi. (2.186)

For u ∈ W 1,2(Ω), we have |u|q−2 ∈ L2∗/(q−2)(Ω) so that, in general, we do not have|u|q−2∇u ∈ L2(Ω) guaranteed. Likewise, the a0- and c0-terms also do not live inL2(Ω) in general if we do not have some additional information about u ∈ W 1,2(Ω).However, we can use the already proved assertion (i), i.e. u ∈ W 2,2

loc (Ω); this trick

is called a bootstrap91. Then it is easy to show that g1 ∈ W 1,2loc (Ω) hence we can

use simply Proposition 2.102 with g being now g1. �

Having the data qualification A ∈ C1(Ω;Rn×n) and a0 ∈ W 1,∞(R;Rn) as-sumed and the W 2,2

loc (Ω)-regularity at our disposal, it is then straightforward tocheck that (2.184) holds not only in the weak sense but even a.e. in Ω. Such amode of a solution to a differential equation is called a Caratheodory solution.

Let us now briefly outline how regularity up to the boundary can be obtained.We will confine ourselves to W 2,2-regularity and the Newton boundary conditions(2.48) and begin with (2.169). Thus (2.48) reads as

n∑j=1

νiaij(x)∂u

∂xj+ b(x, u) = h(x) on Γ. (2.187)

Proposition 2.104 (W 2,2-regularity up to boundary). Let Ω be of C2-class,A∈C1(Ω;Rn×n) satisfy (2.170), b∈C1(Rn×R) satisfy, for some b0>0 and C∈R,

∀(a.a.)x∈Γ ∀r1, r2∈R :(b(x, r1)−b(x, r2)

)(r1−r2) ≥ b0

∣∣r1−r2∣∣2, (2.188a)

∃γ∈L2(Γ) ∀(a.a.)x∈Γ ∀r∈R :∣∣∣ ∂b∂x

(x, r)∣∣∣ ≤ γ(x) + C|r|2#/2, (2.188b)

g∈L2(Ω), h∈W 1,2(Γ),92 and let u∈W 1,2(Ω) be the unique weak solution to theboundary-value problem (2.169)–(2.187). Then u∈W 2,2(Ω). Moreover, if b(x, r) =

b1(x)r with b1 ∈ W 1,2#2/(2#−2)(Γ), then

‖u‖W 2,2(Ω) ≤ C(‖g‖L2(Ω) + ‖h‖W 1,2(Γ)

)(2.189)

with C = C(Ω, ‖A‖C1(Ω;Rn×n), ‖b1‖W 1,2n−2+ε(Γ)

).

91Often, bootstrap is used not only in the order of differentiation but rather in the integrability,which is not possible here because we present the Hilbertian theory only.

92The notation W 1,2(Γ) for Γ smooth means that, after a local rectification like on Figure 8,the transformed and “smoothly cut” functions belong to W 1,2(Rn−1). Also ∂

∂xb in (2.188b) refers

to the derivatives in the tangential directions only.


Sketch of the proof. First, as Ω is bounded, Γ is a compact set in Rn, and canbe covered by a finite number of open sets which are C2-diffeomorphical imagesof the unit ball B = {ξ ∈ Rn; |ξ| ≤ 1} such that the respective part of Γ is animage of {ξ = (ξ1, . . . , ξn) ∈ B; ξ1 = 0}. Thus we rectified locally the boundaryΓ, cf. Figure 8.

��

��

��

Ω

ψdiffeomorphism

x1

x2

ξ1

ξ2

Figure 8. Illustration of finite coverage of Γ ⊂ R2 and one diffeomorphism rectifyinglocally a part of Γ.

It is a technical calculation showing that u ∈ W 1,2(B0) defined by u(ξ) = u(ψ(ξ)),where ψ : B0 := {ξ ∈ B; x1 ≤ 0} → Ω is the homeomorphism in question,is a weak solution to an equation like (2.169) but with the coefficients A trans-formed but again being continuously differentiable and satisfying (2.170)93, andthe boundary condition (2.187) transforms to a similar condition for u|ξ1=0. Hence,in fact, it suffices to obtain an estimate like (2.189) only for u ∈ W 1,2(B0). Forsimplicity, we will use the original notation.

We again use the test function (2.177) but now only for k = 2, . . . , n, i.e. weuse shifts only in the tangential direction, so that we still have (2.178) at ourdisposal. Now the cut-off function ζ : B0 → [0, 1] can be taken as 1 in a semi-ball{ξ ∈ Rn; |ξ| ≤ 1 − ε0, x1 ≤ 0} and vanishing on {ξ ∈ Rn; |ξ| ≥ 1 − 1

2ε0, x1 ≤0} with some ε0. The heuristical estimate (2.173)–(2.174) now involves also theboundary term

∫Γ(b(x, u) − h) ∂

∂xk(ζ2 ∂

∂xku) dS which, in the difference variant,

reads and, for |ε| ≤ 12ε0, can be estimated as∫

Γ

(b(x, u)−h

)D−ε

k

(ζ2Dε

ku)dS = −

∫Γ

ζ2Dεk

(b(x, u)−h

)Dε

ku dS

= −∫Γ

ζ2(b(x+εek, u(x+εek)

)− b(x+εek, u(x)

)ε

−Dεkh+

1

ε

∫ ε

0

∂b

∂r

(x+τek, u(x)

)dτ

)Dε

ku dS

≤ −b0∥∥ζDε

ku∥∥2L2(Γ)

+(∥∥h∥∥

W 1,2(Γ)+∥∥γ+C|u|2#/2

∥∥L2(Γ)

)∥∥ζDεku∥∥L2(Γ)

93To be more specific, u satisfies∑n

i,j=1 ∂(aij∂u/∂xj)∂xi = g with the transformed coefficients

aij(ξ) =∑n

k,l=1[akl∂

∂xkψ−1 ∂

∂xlψ−1](ψ(ξ)) and g(ξ) = g(ψ(ξ)). The boundary conditions are

transformed accordingly, i.e. b(ξ, r) = b(ψ(ξ), r) and h(ξ) = h(ψ(ξ)).


≤ ∥∥h∥∥2W 1,2(Γ)

+1

b0

(∥∥γ∥∥2L2(Γ)

+ C2‖u∥∥2#L2#(Γ)

). (2.190)

In this way, we get the local estimates for ∂2

∂xi∂xju for all (i, j) except i = 1 = j

meant in the locally rectified coordinate system, cf. Figure 8(right).The estimate of the normal derivative follows just from the equation itself

which has been shown to hold a.e. in Ω. Thus

∂2u

∂x21

=1

a11

(g −

∑i+j>2

aij∂2u

∂xi∂xj−

n∑i,j=1

∂aij∂xi

∂u

∂xj

)(2.191)

from which we get the local L2-bound for ∂2

∂x21u near the boundary because a−1

11 ∈L∞(Ω) due to the uniform ellipticity of A.

For the special case b(x, r) = b1(x)r, the estimate (2.190) can be finalized by‖[ ∂

∂xb](x, u)‖L2(Γ) ≤ ‖ ∂∂xb1‖L2#2/(2#−2)(Γ)‖u‖L2#(Γ). This eventually allows us to

derive the a-priori estimate (2.189) by summing the (finite number of) the localestimates on the boundary with one estimate on an open set O from Proposi-tion 2.101 and by using the conventional energy estimate ‖u‖W 1,2(Ω) and thus also‖u‖L2#(Γ) in terms of g and h. �

Corollary 2.105 (W 2,2-regularity for semilinear equation). Let the as-sumptions of Propositions 2.103(i) and 2.104 be satisfied. Then any weak solutionu to the equation (2.184) with the boundary conditions

n∑j=1

νi

(aij(x)

∂u

∂xj+ a0i(u)

)+ b(x, u) = h(x) on Γ (2.192)

is a Caratheodory solution and belongs also to W 2,2(Ω).

Remark 2.106 (Dirichlet boundary conditions). Alternatively, instead of (2.192),one can think about prescribing u|Γ = uD with uD = w|Γ for some w ∈ W 2,2(Ω).After a shift by w, cf. Proposition 2.27, one gets a problem for u0 = u − w withzero Dirichlet condition and a contribution to the right-hand side which is againin L2(Ω). The proof of Proposition 2.104 is even simpler because (2.190) simplyvanishes.

2.8 Bibliographical remarks

Pseudomonotone mappings have been introduced by Brezis [64].94 A further read-ing can involve the books by Necas [305], Pascali and Sburlan [325], Renardy and

94In fact, [64] allows for A : V → V2 with V2 “in duality” with V but not necessarily V2 = V ∗,and also requires u �→ 〈A(u), u − v〉 to be lower bounded on each compact set in V and foreach v ∈ V , which is weaker than (2.3a). In literature, “pseudomonotone” sometimes omit (2.3a)completely, cf. [427, Definition 27.5].

2.8. Bibliographical remarks 93

Rogers [349], Ruzicka [376], and Zeidler [427, Chap.27]. Mere monotone map-pings can be found there, too, and also in a lot of further monographs, say[95, 168, 414, 424]. Historically, theory of monotone mappings arises by the worksby Browder [73], Minty [286], and Vishik [416].

The mappings weakly continuous when restricted to finite-dimensional sub-spaces and satisfying (2.123) are called mappings of the type (M), having been in-vented by Brezis [64], and further generalized e.g. in [201, 227]. This class involvesboth the pseudomonotone and the weakly continuous mappings95 but, contraryto those two classes, it is not closed under addition. Mappings of type (M) donot inherit some other nice properties of pseudomonotone mappings, too.96 Asto the weakly continuous mappings, their importance in the context of semilinearequations has been pointed out by Francu [149]. The setting A : V → Z∗ ⊃ V ∗

with Vk ⊂ Z we used in Section 2.5 was used by Hess [201] in the context of themappings of the type (M), see also [325, Ch.IV, Sect.3.1] or [427, Sect.27.7]. Themappings satisfying (2.23) are called mappings of the type (S+); this notion hasbeen invented by Browder [76, p.279].

The fruitful Galerkin method originated at the beginning of 20th century[171], being motivated by engineering applications.

Concrete quasilinear partial differential equations in the divergence formhas been scrutinized, e.g., by Chen and Wu [92, Chap.5], Fucık and Kufner[159], Gilbarg and Trudinger [178, Chap.11], Ladyzhenskaya and Uraltseva [250,Chap.4], Lions [261, Sect.2.2], Necas [305], Taylor [402, Chap.14], and Zeidler [427,Chap.27]. For semilinear equations see Pao [324]. Quasilinear equations in a non-divergence form (not mentioned in here) can be found, e.g., in Ladyzhenskaya andUraltseva [250, Chap.6] or Gilbarg and Trudinger [178, Chap.12]. Fully nonlin-ear equations of the type a(Δu) = g (also not mentioned in here) are, e.g., inChen, Wu [92, Chap.7], Caffarelli, Chabre [86], Dong [126, Chap.9,10], Gilbargand Trudinger [178, Chap.17].

Regularity results in Sect. 2.7 can easily be generalized for the strongly mono-tone quasilinear equation of the type (2.147) satisfying (2.159a). More general reg-ularity theory for elliptic equations is exposed, e.g., in the monographs by Bensous-san, Frehse [50], Evans [138], Giaquinta [175], Gilbarg, Trudinger [178], Grisvard[190], Lions, Magenes [262], Ladyzhenskaya, Uraltseva [250], Necas [302, 305], Re-nardy, Rogers [349], Skrypnik [387], and Taylor [402]. Besides, this active researcharea is recorded in thousands of papers; e.g. Agmon, Douglis, and Nirenberg [6]and Necas [303].

95For the implication “pseudomonotone⇒ type-(M)” see Exercise 2.54 while the implication“weakly continuous⇒ type-(M)” is obvious – note that even lim supk→∞

⟨A(uk), uk

⟩ ≤ 〈f, u〉occurring in (2.123) does not need to have a sense if A(uk) ∈ Z∗ \ V ∗ or f ∈ Z∗ \ V ∗.

96E.g., Φ′ of type (M) does not yield weak lower-semicontinuity of Φ, unlike pseudomonotonic-ity, cf. Theorem 4.4(ii); e.g. Φ(u) = −‖u‖2 if V is an infinite-dimensional Hilbert space.

Chapter 3

Accretive mappings

Besides bounded mappings from a Sobolev space to its dual, there is an alternativeunderstanding of differential operators as unbounded operators from a (typicallydense) subset of a function space to itself. This calls for a generalization of a mono-tonicity concept for mappings D → X , with X a Banach space and D its subset.Moreover, X need not be reflexive because the weak-compactness arguments willbe replaced by metric properties and completeness. The main benefit from thisapproach will be achieved for evolution problems in Chapter 9 but the method isof some interest in steady-state problems themselves.

3.1 Abstract theory

For brevity, let us agree to write ‖ · ‖ and ‖ · ‖∗ instead of ‖ · ‖X and ‖ · ‖X∗ ,respectively.

Definition 3.1. A duality mapping (in general set-valued) J : X ⇒ X∗ is definedby:

J(u) :={f ∈X∗; 〈f, u〉 = ‖u‖2 = ‖f‖2∗

}. (3.1)

Lemma 3.2. Let X be a separable1 Banach space.

(i) J(u) is nonempty, closed, and convex, and J is (norm,weak*)-upper semicon-tinuous.

(ii) If X∗ is strictly convex, then J is single-valued, demicontinuous (i.e. here(norm,weak*)-continuous), and d-monotone with d : R → R linear.

(iii) If X∗ is uniformly convex, then J is continuous.

(iv) If X is strictly convex, then J is also strictly monotone.

1In fact, if general-topology tools and Alaoglu-Bourbaki’s theorem would be used instead ofBanach’s Theorem 1.7, non-separable spaces can be considered, as well.


96 Chapter 3. Accretive mappings

Proof. (i) Closedness of the set J(u) is obvious while its convexity follows fromthe chain of estimates:

‖u‖2 =⟨12f1 +

1

2f2, u

⟩≤∥∥∥12f1 +

1

2f2

∥∥∥∗‖u‖ ≤

(12‖f1‖∗ + 1

2‖f2‖∗

)‖u‖ = ‖u‖2.

Nonemptyness of J(u) is a consequence of the Hahn-Banach Theorem 1.5, allowingus to separate any v ∈ X , ‖v‖ = 1, from the interior of the unit ball, i.e. there isg ∈ X∗ such that ‖g‖∗ = sup‖u‖=1〈g, u〉 = 1 and 〈g, v〉 = 1. For arbitrary 0 �= u ∈X , we then have f = g‖u‖ ∈ J(u) with g selected as previously for v := u/‖u‖because obviously 〈f, u〉 = ‖u‖〈g, u〉 = ‖u‖2〈g, u/‖u‖〉 = ‖u‖2〈g, v〉 = ‖u‖2 and‖f‖∗ = ‖g‖∗‖u‖ = ‖u‖. If u = 0, then obviously J(u) � 0.

To show the (norm,weak*)-upper semicontinuity of J , take uk → u, fk∗⇀ f ,

and fk ∈ J(uk). Then

〈f, u〉 ← 〈fk, uk〉 = ‖uk‖2 → ‖u‖2 (3.2)

and therefore 〈f, u〉 = ‖u‖2. Since ‖ · ‖∗ is convex and continuous, it is weakly*lower semicontinuous and thus

‖f‖∗ ≤ lim infk→∞

‖fk‖∗ = lim infk→∞

‖uk‖ = limk→∞

‖uk‖ = ‖u‖ =〈f, u〉‖u‖ ≤ ‖f‖∗, (3.3)

where 〈f, u〉 = ‖u‖2 has been used. Altogether, ‖f‖∗ = ‖u‖ and thus f ∈ J(u).(ii) Strict convexity of X∗ and convexity of J(u) implies that J(u) is a sin-

gleton because J(u) always belongs to a sphere in X∗ of the radius ‖u‖. Thedemicontinuity in the sense of the (norm,weak*)-continuity of J then follows fromthe point (i).

The d-monotonicity of J follows from the estimate

〈J(u)− J(v), u − v〉 = ‖J(u)‖∗‖u‖+ ‖J(v)‖∗‖v‖ − 〈J(u), v〉 − 〈J(v), u〉≥ ‖J(u)‖∗‖u‖+ ‖J(v)‖∗‖v‖ − ‖J(u)‖∗‖v‖ − ‖J(v)‖∗‖u‖=(‖J(u)‖∗ − ‖J(v)‖∗

)(‖u‖ − ‖v‖) = (‖u‖ − ‖v‖)2. (3.4)

(iii) Besides J(uk)∗⇀ J(u) for uk → u, we have also ‖J(uk)‖∗ = ‖uk‖ →

‖u‖ = ‖J(u)‖∗ so that Theorem 1.2 yields J(uk) → J(u).(iv) Suppose 〈J(u)− J(v), u− v〉 = 0. From (3.4) immediately follows ‖u‖ =

‖v‖. Suppose, for a moment, that u �= v. Then also u/‖u‖ �= v/‖v‖ and thus‖J(u)‖∗ = 〈J(u), u/‖u‖〉 > 〈J(u), v/‖v‖〉 because the supremum in ‖J(u)‖∗ =sup‖z‖≤1〈J(u), z〉 can be attained in at most one point becauseX is strictly convex;this point is z = u/‖u‖. Therefore 〈J(u), u〉 > 〈J(u), v〉. Similarly, also 〈J(v), v〉 >〈J(v), u〉. Thus

〈J(u)− J(v), u − v〉 = 〈J(u), u〉+ 〈J(v), v〉 − 〈J(u), v〉 − 〈J(v), u〉> 〈J(u), v〉+ 〈J(v), u〉 − 〈J(u), v〉 − 〈J(v), u〉 = 0, (3.5)

a contradiction to 〈J(u)− J(v), u − v〉 = 0. �

3.1. Abstract theory 97

Corollary 3.3. Let X be reflexive and both X and X∗ be strictly convex.2 Then J−1

exists, and is the duality mapping X∗→X. In particular, J−1 is demicontinuous.

Proof. We use Browder-Minty’s theorem 2.18. Thus it remains to show the coer-civity of J : obviously 〈J(u), u〉/‖u‖ = ‖u‖2/‖u‖ = ‖u‖ → +∞ for ‖u‖ → ∞. Bysymmetry of the definition (3.1), J−1 : X∗ → X is then the duality mapping, andby Lemma 3.2(ii) it is demicontinuous. �

Definition 3.4. The mapping A : dom(A)→X , dom(A)⊂X , is called accretive iff

∀u, v∈dom(A) ∃f ∈J(u− v) : 〈f,A(u)−A(v)〉 ≥ 0. (3.6)

If, in addition, I+A is surjective, A is called m-accretive.3

Remark 3.5. Introducing the notation of a so-called semi-inner product 〈·, ·〉s by

〈u, v〉s := sup⟨u, J(v)

⟩, (3.7)

the definition (3.6) can equivalently be written4 as 〈A(u)−A(v), u − v〉s ≥ 0.

Remark 3.6. If −A is accretive (resp. m-accretive), A is called dissipative (resp. m-dissipative).

Lemma 3.7 (Metric properties). The mapping A is accretive if and only if(I+ λA)−1, defined on Range(I+ λA), is non-expansive for any λ > 0, i.e.

‖u− v‖ ≤ ‖u+ λA(u) − v − λA(v)‖. (3.8)

Proof. The “only if” part: Let u, v ∈ dom(A) and 〈f,A(u) − A(v)〉 ≥ 0 for somef ∈ J(u− v). Then

‖u− v‖2 = 〈f, u− v〉 ≤ ⟨f, u− v + λ(A(u)−A(v))⟩

≤ ‖f‖∗∥∥u−v + λ(A(u)−A(v))

∥∥ = ‖u−v‖∥∥u−v + λ(A(u)−A(v))∥∥ (3.9)

from which (3.8) follows for any λ > 0.The “if” part: Conversely, suppose that (3.8) holds. Let fλ ∈ J(u − v +

λ(A(u) − A(v))). The case u = v is trivial because f = 0 ∈ J(0) = J(u − v)satisfies 〈f,A(u)−A(v)〉 ≥ 0. Hence we may assume u �= v. Then fλ �= 0 because,by (3.1) and (3.8), ‖fλ‖∗ = ‖u+ λA(u)− v − λA(v)‖ ≥ ‖u− v‖ > 0 and thus we

2In fact, the strict convexity is not restrictive in this case because, by Asplund’s theorem,every reflexive space can be suitably re-normed so that the new norm is equivalent with theoriginal one and both X and X∗ are strictly convex.

3Sometimes, this is called “hypermaximal accretive” or “hyperaccretive”, cf. Browder [76]Crandal and Pazy [109] or Deimling [118, Sect.13]. For “hyperdissipative” see Yosida [425,Sect.XIV.6].

4This equivalence is due to the weak* compactness of J(v), cf. Lemma 3.2(i) and we realizethat J(v) is certainly bounded, so that the supremum in (3.7) is certainly attained.


can put gλ = fλ/‖fλ‖∗. Then, up to a subsequence, gλ ⇀ g weakly* for λ → 0.Also, by (3.8), for all λ > 0,

‖u−v‖ ≤ ‖u− v + λA(u)− λA(v)‖ = 〈gλ, u− v + λA(u)− λA(v)〉≤ ‖gλ‖∗‖u−v‖+ λ〈gλ, A(u)−A(v)〉 = ‖u−v‖ + λ〈gλ, A(u)−A(v)〉, (3.10)

from which it follows that 〈gλ, A(u)−A(v)〉 ≥ 0 for all λ > 0. For λ → 0, one gets〈g,A(u)−A(v)〉 ≥ 0. By the first part of (3.10), we have also

‖u− v‖ ≤ 〈gλ, u− v + λA(u) − λA(v)〉 → 〈g, u− v〉 ≤ ‖g‖∗‖u− v‖. (3.11)

In particular, (3.11) implies ‖g‖∗ ≥ 1. Since ‖g‖∗ ≤ 1 (because ‖ · ‖∗ is weakly*lower semicontinuous), we get ‖g‖∗ = 1 and then, again from (3.11), we get ‖u−v‖ = 〈g, u− v〉. This shows that f := g‖u− v‖ ∈ J(u− v) from which (3.6) followswith f = g‖u− v‖. �

Proposition 3.8 (m-accretivity). If A is m-accretive, then I + λA is surjectivefor all λ > 0.

Proof. Let f ∈ X and λ > 0. Then u+ λA(u) = f means just

u = (I+A)−1( 1λf +

(1− 1

λ

)u)=: Bλ,A(u) (3.12)

because Range(I + A) = X , cf. also Exercise 3.33. Moreover, (I + A)−1 is non-expansive on X (see Lemma 3.7) thus Bλ,A is Lipschitz continuous with the con-stant |1 − 1

λ |. By the Banach fixed-point Theorem 1.12, (3.12) has a solutionprovided λ > 1

2 . Since f was arbitrary, we may conclude that Range(I+λA) = Xfor λ > 1

2 . For λ ≤ 12 we just iterate this procedure [−log2(λ)]-times, with [·]

denoting here the integer part, relying on the fact that u should also be a fixedpoint of Bλ1,λ2A with λ = λ1λ2 and that, if λ1 > 1

2 , Bλ1,λ2A is non-expansiveprovided we already have proved the surjectivity of I+ λ2A and non-expansivityof its inverse. �

Remark 3.9 (Maximal accretivity). If A is m-accretive then it is maximal accretivewith respect to the ordering of graphs by inclusion.5

Remark 3.10 (Special case dom(A) = X ≡ X∗ Hilbert). If X is a Hilbert space,then J is linear.6 If also X ≡ X∗, then simply J(u) = u. If also dom(A) = X ,

5Cf. Exercise 3.37 below. The opposite implication holds if X is a Hilbert space. In general, itdoes not hold, as shown by Crandall and Liggett [108]. The latter property equivalently meansthat, if A is accretive and if, for all v ∈ dom(A), 〈f − A(v), u − v〉s ≥ 0, then u ∈ dom(A) andA(u) = f .

6In this case, one can define the linear operator J : V → V ∗ by 〈Ju, v〉 := (u, v) with (·, ·)denoting the inner product in V . Then 〈Ju, u〉 = (u, u) = ‖u‖2 and ‖Ju‖∗ = sup‖v‖=1〈Ju, v〉 =sup‖v‖=1(u, v) = ‖u‖, which obviously coincides with the definition (3.1).

3.2. Applications to boundary-value problems 99

then monotonicity just coincides with accretiveness and, moreover, any accretiveradially-continuous A is m-accretive.7

Remark 3.11 (Generalized solutions of u + λA(u) = f). If V is a normed linearspace such that V ⊂ X densely8 and A, defined on V , is m-accretive, by (3.8) wecan extend the uniformly continuous mapping (I + λA)−1 : V → V on X . Thisgives a generalized solution to u + λA(u) = f for f ∈ X , cf. Remark 3.18 below.Sometimes, this equation can be suitably interpreted, cf. (3.33) below.

Remark 3.12 (Solutions of A(u) = f). If A is m-accretive, by quite sophisti-cated arguments, it can be shown that not only A+ I/λ but A itself is surjectiveprovided also X and X∗ are uniformly convex and A is coercive in the senselim‖v‖→∞ ‖A(v)‖ = ∞; cf. Deimling [118, Thm.13.4] or Hu and Papageorgiou[209, Part I, Thm.III.7.48].

3.2 Applications to boundary-value problems

This section, although having an interest of its own, is rather preparatory forChapter 9 and will mainly be exploited there.

3.2.1 Duality mappings in Lebesgue and Sobolev spaces

Let us emphasize that we assume Rm to be endowed by the Euclidean norm | · |so that r · r = |r|2 for r ∈ Rm. For another choice, see Exercise 3.39 below.

Proposition 3.13 (Duality mapping for Lebesgue spaces). Let X =Lp(Ω;Rm). If 1 < p < +∞, then J : Lp(Ω;Rm) → Lp′

(Ω;Rm) is given by

J(u)(x) =u(x)|u(x)|p−2

‖u‖p−2Lp(Ω;Rm)

(3.13)

for a.a. x ∈ Ω. For p = 1, J is a set-valued mapping given by

J(u) = ‖u‖L1(Ω;Rm)Dir(u) where

Dir(u) :={f ∈L∞(Ω;Rm); f(x)∈dir

(u(x)

)for a.a. x∈Ω

}, (3.14)

where “dir” denotes the direction of the vector indicated, i.e.

dir(r) :=

{r/|r| if r �= 0,{r∈Rm; ‖r| ≤ 1

}if r = 0.

(3.15)

7Indeed, I : X → X ≡ X∗ is monotone, bounded, and radially continuous and also coercive,

hence so is I + A; the coercivity of I + A follows from〈u+A(u),u〉

‖u‖ = ‖u‖ + 〈A(u),u〉‖u‖ → ∞ for

‖u‖ → ∞ because, by monotonicity, the term 〈A(u), u〉 has at most linear decay: 〈A(u), u〉 =〈A(u)−A(0), u− 0〉+ 〈A(0), u〉 ≥ −‖A(0)‖∗‖u‖. Then, by Browder-Minty’s Theorem 2.18, A issurjective.

8Then X is a so-called completion of V (with respect to a uniformity induced by the norm).


Proof. Recalling that Lp′(Ω;Rm) is uniformly convex, see Section 1.2.2, by

Lemma 3.2(ii), J(u) has just one element, and we are to verify (3.13). Indeed,we obviously have

⟨J(u), u

⟩=

∫Ωu(x)|u(x)|p−2 · u(x) dx

‖u‖p−2Lp(Ω;Rm)

= ‖u‖2−pLp(Ω;Rm)

∫Ω

|u|pdx = ‖u‖2Lp(Ω;Rm)

and also

∥∥J(u)∥∥p′

Lp′(Ω;Rm)=

∫Ω

∣∣∣u(x)∣∣u(x)∣∣p−2∥∥u∥∥2−p

Lp(Ω;Rm)

∣∣∣p′

dx

=

∫Ω

∣∣u(x)∣∣p∥∥u∥∥(2−p)p/(p−1)

Lp(Ω;Rm)dx=

∥∥u∥∥pLp(Ω;Rm)

∥∥u∥∥(2−p)p/(p−1)

Lp(Ω;Rm)=∥∥u∥∥p′

Lp(Ω;Rm).

As to the case p = 1, we obviously have

⟨J(u), u

⟩=∥∥u∥∥

L1(Ω;Rm)

∫Ω

f(x) · u(x) dx =∥∥u∥∥2

L1(Ω;Rm)(3.16)

for any f ∈ Dir(u); note that always u(x) · dir(u(x)) = |u(x)|. Also, if u �= 0,∥∥J(u)∥∥L∞(Ω;Rm)

=∥∥u∥∥

L1(Ω;Rm)ess supx∈Ω

∣∣dir(u(x))∣∣ = ∥∥u∥∥L1(Ω;Rm)

. (3.17)

If u = 0, then the desired equality ‖J(u)‖L∞(Ω;Rm) = 0 = ‖u‖L1(Ω;Rm) holds, too.This proved the inclusion “⊃” in (3.14). The opposite inclusion follows by a moredetailed analysis of the above formulae. �

Proposition 3.14 (Duality mapping for Sobolev spaces). Let X = W 1,p0 (Ω),

1 < p < +∞, normed by ‖u‖W 1,p0 (Ω) = ‖∇u‖Lp(Ω;Rn). Then

J(u) = − Δpu

‖u‖p−2

W 1,p0 (Ω)

. (3.18)

Proof. Uniform convexity of Lp′(Ω;Rm) makes also W 1,p

0 (Ω)∗ uniformly convex.Hence, by Lemma 3.2(ii) J(u) has just one element, and we are to verify (3.18).By Green’s formula (1.54), we indeed have

⟨J(u), u

⟩=

〈−Δpu, u〉‖u‖p−2

W 1,p0 (Ω)

=

∫Ω |∇u|p−2∇u · ∇u dx

‖u‖p−2

W 1,p0 (Ω)

= ‖u‖2W 1,p

0 (Ω)(3.19)

and, using again Green’s formula (1.54) and Holder’s inequality, also


∥∥J(u)∥∥W 1,p

0 (Ω)∗ = sup‖v‖

W1,p0 (Ω)

≤1

〈Δpu, v〉‖u‖p−2

W 1,p0 (Ω)

= sup‖v‖

W1,p0 (Ω)

≤1

∫Ω|∇u|p−2∇u · ∇v dx

‖u‖p−2

W 1,p0 (Ω)

≤ sup‖v‖

W1,p0

(Ω)≤1

‖∇u‖p−1Lp(Ω;Rn)‖∇v‖Lp(Ω;Rn)

‖u‖p−2

W 1,p0 (Ω)

= ‖u‖W 1,p0 (Ω) (3.20)

and the supremum (and thus equality) is attained for v = u/‖u‖W 1,p0 (Ω). �

Remark 3.15. For X = W 1,p(Ω) normed by ‖u‖W 1,p(Ω) = (‖∇u‖pLp(Ω;Rn) +

‖u‖pLp(Ω))1/p, we have

J(u) =|u|p−2u−Δpu

‖u‖p−2W 1,p(Ω)

. (3.21)

Note that we used (3.21) already for p = 2 in (2.161).

3.2.2 Accretivity of monotone quasilinear mappings

Let us consider A corresponding to (2.147), i.e. A(u) = −div a(x,∇u) + c(x, u),and investigate its accretivity in X = Lq(Ω); for this we define9

dom(A) :={u∈W 1,p(Ω); div a(x,∇u)− c(x, u)∈Lq(Ω), u|ΓD = uD,

ν · a(x,∇u) + b(x, u) = h(x) on ΓN in the “weak sense”}

(3.22)

with h∈Lp#′(ΓN) and uD satisfying (2.58) fixed. As in Exercise 2.88, we assume

here (2.148), i.e. in particular that a(x, ·), b(x, ·), and c(x, ·) are monotone.

Proposition 3.16 (Accretivity). Let 1 ≤ q ≤ p∗, q < +∞, and let (2.148),(2.57) concerning h, and (2.58) hold. Then A(u) = −div a(x,∇u) + c(x, u) posedas (3.22) is accretive on Lq(Ω).

Proof. First, note that dom(A) ⊂ W 1,p(Ω) ⊂ Lq(Ω) =: X provided q ≤ p∗.If q = 1, then “dir” (which is just “sign” form = 1) in (3.15) is not continuous

at 0, and we must use its regularization signε : R → [−1, 1] defined, e.g., by

signε(r) =

{r/|r| if |r| ≥ ε,

ε−1r otherwise.(3.23)

9For q ≥ p∗′, (3.22) means that we take all weak solutions u ∈ W 1,p(Ω) for the boundary-value problem (2.147) with some g ∈ Lq(Ω) and with the boundary condition specified.


Then, for u, v ∈ dom(A) and for f = sign(u− v) ∈ J(u− v)/‖u− v‖L1(Ω), we have

⟨f,A(u)−A(v)

⟩=

∫Ω

sign(u−v)(c(u)−c(v)− div

(a(∇u)−a(∇v)

))dx

= limε→0

∫Ω

signε(u−v)(c(u)−c(v) − div

(a(∇u)−a(∇v)

))dx

= limε→0

∫Ω

(a(∇u)−a(∇v)

)·∇signε(u−v) +(c(u)−c(v)

)signε(u−v) dx

+

∫ΓN

(b(u)− b(v)

)signε(u − v) dS ≥ 0 (3.24)

where we used, beside Green’s formula (1.54), the convergence

limε→0

signε(u(x) − v(x)) = sign(u(x) − v(x)) =

⎧⎨⎩1 if u(x) > v(x),0 if u(x) = v(x),−1 if u(x) < v(x),

(3.25)

for a.a. x ∈Ω and then we used the Lebesgue Theorem 1.14 with the integrablemajorant |div(a(∇u)− a(∇v)) − c(u) + c(v)| ∈ L1(Ω). The inequality in (3.24) isbecause (a(∇u)− a(∇v)) · ∇signε(u− v) = (a(∇u)− a(∇v))(∇u −∇v)sign′ε(u−v) ≥ 0, cf. Proposition 1.28, and also (c(u) − c(v))signε(u−v) ≥ 0, and similarly(b(u)− b(v))signε(u−v) ≥ 0.

For q > 1, we use J from (3.13). As the case u = v is trivial, let us take u, v ∈dom(A), u �= v, and define ωq(r) := r|r|q−2/‖u− v‖q−2

Lq(Ω). Besides, let us consider

a Lipschitz continuous regularization ωq,ε of ωq such that limε→0 ωq,ε(r) = ωq(r)for all r and ωq,ε(r) ≤ ωq(r) for r ≥ 0 and ωq,ε(r) ≥ ωq(r) for r ≤ 0. By usingLebesgue’s Theorem 1.14 and Green’s formula (1.54), we can calculate10

〈J(u−v), A(u)−A(v)〉 =∫Ω

ωq(u−v)(c(u)−c(v)− div

(a(∇u)−a(∇v)

))dx

= limε→0

∫Ω

ωq,ε(u− v)(c(u)− c(v)− div

(a(∇u)− a(∇v)

))dx

= limε→0

(∫Ω

(a(∇u)− a(∇v)

) · ∇ωq,ε(u−v) dx

+

∫Ω

(c(u)−c(v)

)ωq,ε(u−v) dx +

∫ΓN

(b(u)−b(v)

)ωq,ε(u−v) dS

)=: lim

ε→0

(I1,ε + I2,ε + I3,ε

). (3.26)

10Note that (u−v)|u−v|q−2 ∈ Lq′ (Ω) if q ≤ p∗ while div(a(∇u)−a(∇v)

)+c(u)−c(v) ∈ Lq(Ω)

so that the product is indeed integrable and, up to a factor ‖u − v‖2−qLq(Ω)

, it also forms the

integrable majorant for the collection {ωq,ε(u−v)(c(u)−c(v)− div(a(∇u)−a(∇v)

))}ε>0 needed

for the limit passage by Lebesgue’s Theorem 1.14.


The first integral I1,ε can be estimated as

I1,ε =

∫Ω

(a(∇u)− a(∇v)

)·∇ωq,ε(u − v) dx

=

∫Ω

(a(∇u)− a(∇v)

)·∇(u− v)ω′q,ε(u− v) dx ≥ 0; (3.27)

note that ωq,ε : R → R is monotone and ∇ωq,ε(u − v) = ω′q,ε(u − v)∇(u − v);cf. Proposition 1.28. Of course, we used the monotonicity of a(x, ·) and that ω′q,εis bounded. The monotonicity of c(x, ·) and of b(x, ·) obviously gives I2,ε ≥ 0 andI3,ε ≥ 0, respectively; the at most linear growth of ωq,ε gives a good sense to bothI2,ε and I3,ε. �

Proposition 3.17 (m-accretivity). Let, beside the assumptions of Proposi-tion 3.16, also q ≥ p∗′. Then A(u) = −div a(x,∇u) + c(x, u) posed as (3.22)is m-accretive on Lq(Ω).

Proof. We are to show that the equation u− div a(∇u) + c(u) = g has a solutionu ∈ dom(A) for any g ∈ X = Lq(Ω). As we assume q ≥ p∗′, we have Lq(Ω) ⊂Lp∗′

(Ω), and there is u ∈ W 1,p(Ω) solving the boundary-value problem (2.147)in the weak sense. Then it suffices to show u ∈ dom(A). Indeed, in the sense ofdistributions it holds that

div a(∇u)− c(u) = u− g ∈ Lq(Ω) (3.28)

because u ∈ W 1,p(Ω) ⊂ Lp∗(Ω) ⊂ Lq(Ω) provided p∗ ≥ q. �

Remark 3.18 (Generalization for q < p∗′). The restriction p∗′ ≤ p∗ we implicitlymade in Propositions 3.16-3.17 requires p ≥ 2n/(n+2) and calls for some sort ofextension if q �∈ [p∗′, p∗] even if this interval is nonempty. For q ∈ [1, p∗′), one canstill perform the estimate 〈A(u) − A(v), u − v〉s ≥ 0 as in (3.24) or (3.26), andthen by (3.9) one can prove the uniform continuity of the mapping (I + A)−1 inthe Lq(Ω)-norm. One can then make an extension by continuity of this mappingto get a generalized solution to div a(∇u) − (c(u) + u) = g ∈ Lq(Ω) with theabove boundary conditions, cf. Remark 3.11. Unlike the distributional solution(3.33) below, more concrete interpretation is not entirely obvious unless one getsadditional information about ∇u.11

Remark 3.19 (The case q = +∞). Investigation of q > p∗ would require to show anadditional regularity of solutions to the boundary-value problem (2.147) to showdom(A) ⊂ Lq(Ω). The case q = +∞, which we avoided in Propositions 3.16-3.17anyhow not to speak about L∞(Ω)∗, is exceptional and can be treated by a special

11A concept of the renormalized and the entropy solutions has been developed for it; seeBenilan et al. [47] and references therein.


comparison technique. Instead of (3.22), let us set12

dom(A) :={u∈W 1,p(Ω) ∩ L∞(Ω); div a(x,∇u)− c(x, u) ∈ L∞(Ω),

u|ΓD = uD, ν · a(x,∇u) + b(x, u) = h on ΓN

}. (3.29)

Let us assume, in addition, that a(x, s) · s ≥ 0, c(x, r)r ≥ 0, h ∈ L∞(Γ), andb(x, ·)−1 does exist and b−1(h) ∈ L∞(Γ).

As the dual space toX = L∞(Ω) is a very abstract object, we avoid specifyingthe duality mapping J : L∞(Ω) → L∞(Ω)∗ and will rather rely on the formula(3.8). For any g ∈ L∞(Ω) and λ > 0, there is a weak solution u ∈ W 1,p(Ω) to theboundary-value problem

u− λdiv a(∇u) + λc(u) = g on Ω,

ν · a(∇u) + b(u) = h on ΓN,

u|ΓD = uD on ΓD.

⎫⎪⎪⎬⎪⎪⎭ (3.30)

Putting G := max(‖uD‖L∞(ΓD), ‖g‖L∞(Ω), ‖b−1(h)‖L∞(Ω)

)and testing the weak

formulation of (3.30)13 by v = (u−G)+, we can see that u ≤ G a.e. in Ω; cf. Ex-ercise 3.36. Likewise, the test by v = (u+G)− yields u ≥ −G a.e. in Ω. Thusu ∈ L∞(Ω) and div a(∇u)−c(u) = (u−g)/λ ∈ L∞(Ω), so that u ∈ dom(A) from(3.29).

Now, take g1, g2 ∈ L∞(Ω) and the corresponding u1, u2 ∈ dom(A), subtractthe corresponding problems (3.30) from each other, and moreover subtract theconstant G = ‖g1− g2‖L∞(Ω) from both sides, i.e. (u1−u2−G)−λ

(div(a(∇u1)−

a(∇u2))− c(u1)+ c(u2)

)= g1− g2−G with the boundary condition ν · (a(∇u1)−

a(∇u2))+ b(u1)− b(u2) = 0 on ΓN. Test the weak formulation with v = (u1−u2−

G)+. Using ∇v = χ{x∈Ω; u1(x)−u2(x)>G}∇(u1−u2), cf. Proposition 1.28, this gives∫Ω

((u1 − u2 −G)+

)2+ λ

(c(u1)− c(u2)

)(u1 − u2 −G)+dx

+

∫{x∈Ω; u1(x)−u2(x)>G}

λ(a(∇u1)− a(∇u2)

)· ∇(u1 − u2) dx

+

∫ΓN

λ(b(u1)−b(u2)

)(u1−u2−G)+dS =

∫Ω

(g1−g2−G)(u1−u2−G)+dx ≤ 0.

The left-hand side can be lower-bounded by ‖(u1 − u2 −G)+‖2L2(Ω), which shows

u1−u2 ≤ G a.e. on Ω. Likewise, testing by v = (u1−u2+G)− yields u1−u2 ≥ −Ga.e. in Ω. Altogether, ‖u1 − u2‖L∞(Ω) ≤ G = ‖g1 − g2‖L∞(Ω) so that the mappingg �→ u is a contraction on X = L∞(Ω).

12In fact, (3.29) coincides with (3.22) for q = +∞ if p > n because then automaticallyW 1,p(Ω) ⊂ L∞(Ω).

13Here we must assume p ≥ 2n/(n+2), so that p∗ ≥ 2 to satisfy |r + λc(r)| ≤ C(1 + |r|p∗−1)like in (2.148c).


Remark 3.20 (Alternative setting). We can define dom(A) more explicitly than(3.22) if a regularity result is employed. E.g., assuming a(x, s) := A(x)s and asmooth data Ω, A, b, and c as in Corollary 2.105, ΓN := Γ, we can define dom(A) ={u ∈ W 2,2(Ω); νA∇u + b(u) = h a.e. on Γ}. The m-accretivity of A : v �→c(v)− div(A∇v) on L2(Ω) then follows from Corollary 2.105.

Example 3.21 (Advection term14). One can modify A from (3.22) by consideringc(x, r, s) := �v(x) · s with a vector field �v : Ω → Rn such that div �v ≤ 0 and(�v|Γ) · ν ≥ 0 as in Exercise 2.91. Again, let q ∈ [p∗′, p∗]. Then, abbreviatingωq(r) := r |r|q−2 and using Green’s formula, the accretivity follows from:∫

Ω

(u−v)|u−v|q−2�v · ∇(u−v) dx =

∫Ω

ωq(u−v)�v · ∇(u−v) dx

=

∫Ω

�v · ∇ωq(u−v) dx =

∫Γ

(�v·ν) ωq(u−v) dS −

∫Ω

(div�v

)ωq(u−v) dx ≥ 0 (3.31)

where ωq is a primitive function of ωq.

3.2.3 Accretivity of heat equation

We will demonstrate the L1-accretive structure of the semilinear heat operator inisotropic media, i.e. u �→ −div(κ(u)∇u), cf. Example 2.79 with B = I.15 Exceptn = 1, the previous approaches do not allow treatment of a heat source of finiteenergy, i.e. bounded only in L1(Ω); cf. also (2.127). Here we put off this restriction,i.e. we take X = L1(Ω). For simplicity, let c be monotone with at most lineargrowth, and let b = 0. Then A(u) := −Δ κ (u) + c(u), and considering Neumannboundary conditions we put

dom(A) :={u∈L1(Ω); Δ κ (u)∈L1(Ω),

∂

∂νκ (u) = h on Γ

}(3.32)

where Δ κ (u) is understood in the sense of distributions and ∂∂ν κ (u) = h is

understood in the weak sense. Then u+A(u) = g for u ∈ dom(A) means preciselythat u ∈ L1(Ω) with Δ κ (u) ∈ L1(Ω) in the sense of distributions and that, byusing Green’s formula (1.54) twice,∫

Ω

uv − κ (u)Δv + c(u)v dx =

∫Ω

gv dx

+

∫Γ

(∂ κ (u)

∂νv − κ (u)

∂v

∂ν

)dS =

∫Ω

gv dx+

∫Γ

hv dS (3.33)

for any v ∈ C∞(Ω) such that ∂∂ν v = 0 on Γ; note that we used u ∈ dom(A)

to substitute ∂∂ν κ (u) = h into the integral on Γ. The important fact is that

14Cf. also Rulla [375].15See Brezis and Strauss [69], or also Barbu [37, Chap. III, Sect. 3.3], Magenes, Verdi, Visintin

[267], Showalter [383, Theorem 9.2].


this set of test functions has sufficiently rich traces on Γ.16 The integral identity(3.33) defines a so-called distributional solution, sometimes also called a very weaksolution.

Proposition 3.22 (m-accretivity). Let c : R → R be monotone with at most

linear growth, 0 < ess inf κ(·) ≤ ess supκ(·) < +∞, and h∈L2#′(Γ). Then A :=

−Δ κ (u) + c(u) with dom(A) from (3.32) is m-accretive on L1(Ω).

Proof. For clarity, we divide the proof into four steps.

Step 1: Considering the weak-solution concept, u + A(u) = g has a solution u ∈dom(A) for any g ∈ L2∗′

(Ω); see Example 2.79 which gives u ∈ W 1,2(Ω) satisfying

∀v∈W 1,2(Ω) :

∫Ω

∇ κ (u)·∇v + uv + c(u)v − gv dx =

∫Γ

hv dS (3.34)

and realize that additionally Δ κ (u) = u+ c(u)− g ∈ L2∗′(Ω) ⊂ L1(Ω) and (3.34)

implies (3.33) for v ∈ C∞(Ω) with ∂∂ν v = 0.

Step 2: We will show that the mapping g �→ u is non-expansive in the L1-norm. Weconsider g := g1 and g2 and the corresponding u1 and u2, write (3.34) for u1 andu2, then subtract; note that κ (u1) and κ (u2) live in W 1,2(Ω), cf. Proposition 1.28.The resulting identity holds not only for v∈D(Ω) but even for v∈W 1,2(Ω). Thuswe can put v = signε( κ (u1) − κ (u2)) ∈ W 1,2(Ω) where signε : R → [−1, 1] wasdefined by (3.23). This results in∫

Ω

(u1 − u2 + c(u1)− c(u2)

)signε

(κ (u1)− κ (u2)

)+∇( κ (u1)− κ (u2)

) · ∇signε(κ (u1)− κ (u2)

)dx

=

∫Ω

(g1 − g2) signε(κ (u1)− κ (u2)

)dx ≤

∫Ω

|g1 − g2| dx. (3.35)

Because of the monotonicity of signε, c, and κ , the term

∇( κ (u1)− κ (u2))·∇signε

(κ (u1)− κ (u2)

)=∣∣∇ κ (u1)−∇ κ (u2)

∣∣2sign′ε( κ (u1)− κ (u2))

is non-negative and also(c(u1)−c(u2)

)signε

(κ (u1)− κ (u2)

) ≥ 0 a.e.; we thus get∫Ω

(u1 − u2) signε(κ (u1)− κ (u2)

)dx ≤ ∥∥g1 − g2

∥∥L1(Ω)

. (3.36)

Realizing that {(u1 − u2)signε( κ (u1) − κ (u2))}ε>0 has an integrable majorant,namely |u1 − u2|, and that this sequence converges to it a.e., we get

limε→0

∫Ω

(u1−u2) signε(κ (u1)− κ (u2)

)dx =

∫Ω

∣∣u1−u2

∣∣ dx =∥∥u1−u2

∥∥L1(Ω)

(3.37)

16Here we use the same arguments as in the proof of Proposition 2.44.


thanks to Lebesgue’s dominated-convergence Theorem 1.14. Joining (3.36) with(3.37) proves g �→ u to be non-expansive in L1(Ω).

Step 3. The limit passage: Take gk ∈ L2∗′(Ω), g ∈ L1(Ω), gk → g in L1(Ω),

uk ∈ dom(A) such that uk+A(uk) = gk. By the Step 2, uk → u in L1(Ω). As κ isLipschitz, also κ (uk) → κ (u) in L1(Ω). As c has at most linear growth, we havealso c(uk) → c(u) in L1(Ω). By Green’s Theorem 1.31 applied to (3.34), we knowthat ∫

Ω

ukv − κ (uk)Δv + c(uk)v dx =

∫Ω

gkv dx+

∫Γ

hv dS (3.38)

which gives in the limit∫Ωuv− κ (u)Δv+c(u)v dx=

∫Ωgv dx+

∫Γhv dS for any v∈

C∞(Ω); ∂∂ν v = 0, i.e. we get (3.33), thus u solves u+A(u) = g with the boundary

condition ∂∂ν κ (u)=h on Γ in the weak sense. Besides, (3.38) implies, in the sense

of distributions, Δ κ (u) = u+c(u)−g, so that Δ κ (u) ∈ L1(Ω). Altogether, u ∈dom(A).

Step 4: The accretivity: By the extension of the estimate in Step 2, we get that(I+A)−1 : g �→ u : L1(Ω) → dom(A) is non-expansive. By the same technique, itcan be proved that also (I + λA)−1 is non-expansive for any λ > 0. From this, Ais accretive, cf. Lemma 3.7. �

Remark 3.23 (Very weak solution to steady-state heat problem). The previousconsiderations can immediately give a very weak solution for the heat equa-tion −div

(κ(u)∇u

)+ c(x, u) = g with g ∈ L1(Ω) and with c strongly monotone

(c(x, r1)−c(x, r2))(r1−r2) ≥ ε(r1−r2)2 so that u−ε−1

(div(κ(u)∇u

)+c0(u)

)= g

with c0(x, r) := c(x, r)−εr is still monotone. In case n = 2, this describes the heat-conductive plate with the convection coefficient c1(x) ≥ ε > 0, cf. (2.132) and Fig-ure 6b. In the case c = 0, which would correspond rather to Figure 6a, the situationis more difficult and Remark 3.12 can apply. Moreover, varying also h and modify-ing (3.35) appropriately, one gets ‖u1−u2‖L1(Ω) ≤ ‖g1−g2‖L1(Ω)+‖h1−h2‖L1(Γ),which allows for extension for h ∈ L1(Γ).

Remark 3.24 (Other boundary conditions). The modification for the Dirichletboundary condition u|Γ = uD is quite technical. In (3.32), instead of the Neumanncondition, one should involve κ (u)|ΓD = uD with uD = κ (uD), and then (3.33)with h = 0 should hold just for v ∈ D(Ω). For the limit passage in Step 3 ofthe above proof, we need also to show that κ (uk)|Γ → κ (u)|Γ at least weakly inL1(Γ). For this, we use boundedness of Δ κ (uk) in L1(Ω), and the deep results ofBoccardo and Gallouet [56, 57], showing that κ (uk) is bounded also in W 1,q(Ω)

with q < n/(n−1), so that κ (uk)|Γ is bounded in Lq#(Γ) with, due to (1.37),q# < (n−1)/(n−2). More precisely, [56, 57] uses zero boundary condition, hencewe must first shift the mapping as in Proposition 2.27 for which we must qualifyuD as being the trace of some w∈W 1,q(Ω) with Δw∈L1(Ω).

For Newton boundary conditions we refer to Benilan, Crandall, Sacks [49].


Remark 3.25 (Heat equation with advection). The mapping A(u) := c(u)�v · ∇u−div(κ(u)∇u), cf. (2.136), allows for L1-accretivity after a so-called enthalpy trans-formation by introducing the new variable w := c (u) where c (r) :=

∫ r

0 c(�) d�is a primitive function to c. Then obviously A(u) = �v · ∇ c (u) − Δ( κ (u)) =�v · ∇w −Δβ(w) with β = κ ◦ [ c ]−1. Then, assuming �v∈W 1,∞(Ω;Rn) such thatdiv�v ≤ 0 and �v|Γ · ν = 0, we can merge the calculations (3.31) with signε in placeof ωq with the arguments in Proposition 3.22 with κ ◦ [ c ]−1 in place of κ . Theidentity (3.33) augments by the term u(�v · ∇v)− u(div�v)v which allows easily fora limit passage as in (3.38); note that the boundary term u(�v · ν) is assumed zerootherwise the limit passage would be doubtful.

3.2.4 Accretivity of some other boundary-value problems

An accretive structure may arise also in a so-called conservation law posed on aone-dimensional domain Ω := (0, 1) as

A(u) =d

dxF (u), X := L1(0, 1), (3.39a)

dom(A) :={u∈W 1,1(0, 1); u(0) = uD,

d

dxF (u)∈L1(0, 1)

}. (3.39b)

Proposition 3.26 (m-accretivity). Let F : R → R be continuous and stronglymonotone. Then A : dom(A) → X defined by (3.39) is m-accretive.

Proof. For the accretivity, we choose f = sign(u−v) ∈ J(u−v)/‖u−v‖L1(0,1).Then:⟨

f,A(u)−A(v)⟩=

∫ 1

0

sign(u− v)d

dx

(F (u)−F (v)

)dx

=

∫ 1

0

d

dx

∣∣F (u)−F (v)∣∣ dx =

∣∣F (u(1))−F (v(1))∣∣ ≥ 0, (3.40)

where we used also that sign(u− v) = sign(F (u)−F (v)) by strict monotonicity ofF , and that u(0) = uD = v(0), and for g = F (u)− F (v) we used also the identity∫ 1

0sign(g) d

dxg dx =∫ 1

0ddx |g| dx, which follows by a regularization technique17. As

to m-accretivity, we are to show that the ordinary differential equation

dw

dx+G(w(x)) = f(x) , w(0) = F (uD), (3.41)

17Using signε : R→ [−1, 1] defined for ε > 0 by (3.23), by Lebesgue’s dominated convergencetheorem, it holds that

|g(1)|ε =

∫ 1

0

d

dx|g|ε dx =

∫ 1

0signε(g)→

∫ 1

0sign(g)

d

dxg dx

because signε(g) → sign(g) a.e. and has an L∞-majorant while ddx

g lives in L1(0, 1). On the

other hand, also |g(1)|ε → |g(1)| = ∫ 10

ddx|g|dx. Note that signε has a convex potential which we

denote by | · |ε, i.e. signε(r) = (|r|ε)′. Moreover, we can suppose |0|ε = 0.


withG = F−1 has a solution. As F has growth at least linear,G has growth at mostlinear, and then existence of w solving (3.41) follows by the standard arguments;cf. Theorem 1.45. For such w, u = G(w) solves u + A(u) = f . Moreover, forf ∈ L1(0, 1) we have w ∈ W 1,1(0, 1). Then d

dxu = ddxG(w) = G′(w) d

dxw ∈ L1(0, 1)providedG′(w) ∈ L∞(0, 1); this requiresG to be Lipschitz for bounded arguments,i.e. F strongly monotone. Then u ∈ W 1,1(0, 1) and also d

dxF (u) = f−u ∈ L1(0, 1)and therefore u ∈ dom(A). �Remark 3.27 (Scalar conservation law on Rn). Assuming F ∈ C1(R;Rn) andlim sup|u|→0 |F (u)|/|u| < +∞, the mapping A, defined as the closure in L1(Rn)×L1(Rn) of the mapping u �→ div(F (u)) : C1

0 (Rn) → C0(Rn), has been shown to be

m-accretive on L1(Ω) in Barbu [38, Section 2.3, Proposition 3.11].

Remark 3.28 (Hamilton-Jacobi equation). For F :R→R increasing and Ω = (0, 1),the m-accretivity of the so-called (one-dimensional) Hamilton-Jacobi operator

A(u) = F(dudx

), X :=

{v∈C([0, 1]); v(0) = 0 = v(1)

}, (3.42a)

dom(A) :={u∈C1([0, 1]); u∈X,

du

dx∈X

}(3.42b)

has been shown in Deimling [118, Example 23.5].

3.2.5 Excursion to equations with measures in right-hand sides

We saw that the accretivity approach allows us sometimes to solve equations withintegrable functions on the right-hand side even if L1(Ω) was not contained inW 1,p(Ω)∗ and the usual a-priori estimates on W 1,p(Ω) fail. Functions from L1(Ω)are special Radon measures on Ω, namely those which are absolutely continuous,and a natural question arises whether one can get rid of this absolute continuityand thus consider general measures in the right-hand sides. Such generalization isoften needed in optimal-control theory of elliptic problems.18

It is indeed possible in special cases but one must always employ quite so-phisticated techniques not necessarily related to the accretivity approach, andoften even a negative answer is known. Here we demonstrate only a rather sim-ple so-called transposition method combined with regularity results, applicable tosome semilinear equations. We will demonstrate it on the Newton-boundary-valueproblem:

−div(A(x)∇u

)+ c(x, u) = μ in Ω,

νA(x)∇u + b1(x)u = η on Γ,

}(3.43)

with μ ∈ M (Ω) and η ∈ M (Γ). We assume n ≤ 3, Ω ⊂ Rn a domain of C2-class, A∈C1(Ω;Rn×n) being uniformly positive definite in the sense of (2.170), b1

18Measures typically occur either in the so-called adjoint systems as Lagrange multiplierson state constraints or in the controlled systems as a result of concentration phenomena if anoptimal-control problem has only an L1-coercivity but not coercivity in Lp for p > 1.


qualified as in Proposition 2.104, i.e. b1(x) ≥ b0 > 0 and b1 ∈ W 1,2#2/(2#−2)(Γ),and c a Caratheodory function satisfying

∃γ0, γ1∈L1(Ω), C∈R+, q <n

n−2, εc > 0 ∀(a.a.)x∈Ω ∀r∈R :

|c(x, r)| ≤ γ0(x) + C|r|q , c(x, r)r ≥ (εc|c(x, r)| − γ1(x)) |r| (3.44)

of course, we mean q < +∞ for n ≤ 2 (while q < 3 for n = 3). We call u ∈ Lq(Ω)a distributional solution to (3.43) if the integral identity obtained like (2.51) butusing Green’s formula twice, i.e.∫

Ω

c(x, u)v − u div(A(x)∇v

)dx =

∫Ω

v μ(dx) +

∫Γ

v η(dS), (3.45)

is valid for all

v ∈ W 2,2(Ω) : νA(x)∇v + b1(x)v = 0 on Γ. (3.46)

Lemma 3.29 (The case c = 0). For any μ ∈ M (Ω) and η ∈ M (Γ), the equa-tion −div(A(x)∇u) = μ with the boundary condition νA(x)∇u + b1(x)u = ηhas a unique distributional solution and the a-priori estimate ‖u‖Wλ,2(Ω) ≤C(‖μ‖M (Ω) + ‖η‖M (Γ)

)holds for λ < 2−n/2 and some C < +∞.

Proof. Let us consider the auxiliary linear problem

−div(A(x)∇v

)= g in Ω,

νA(x)∇v + b1(x)v = 0 on Γ.

}(3.47)

The existence of the weak solution v ∈ W 1,2(Ω) to (3.47) can be proved by thestandard energy method by testing (3.47) by v itself, and we have the estimate‖v‖W 1,2(Ω) ≤ K1‖f‖W 1,2(Ω)∗ with f determined by the pair (g, 0) due to (2.60).As b1 > 0, the solution to (3.47) is unique, and thus defines a linear operatorB : g �→ v. Then we use Proposition 2.104 to claim the W 2,2-regularity for (3.47),i.e. ‖v‖W 2,2(Ω) ≤ K2‖g‖L2(Ω); cf. (2.189).

The interpolation between the linear mappings B : W 1,2(Ω)∗ → W 1,2(Ω)and B : L2(Ω) → W 2,2(Ω) gives a mapping B : Wλ,2(Ω)∗ → W 2−λ,2(Ω) and anestimate ‖v‖W 2−λ,2(Ω) ≤ K‖g‖Wλ,2(Ω)∗ for any λ ∈ [0, 1] and some K dependingon K1, K2, and λ, cf. (1.45).

Let us rewrite the integral identity (3.45)–(3.46) with c ≡ 0, which definesthe distributional solution to the problem considered here, into the form 〈g, u〉 =〈F,Bg〉 for any g = div(A∇v) ∈ L2(Ω) with F defined by

〈F, v〉 =∫Ω

v μ(dx) +

∫Γ

v η(dS). (3.48)

We choose 0 ≤ λ ≤ 1 so small that W 2−λ,2(Ω) ⊂ C(Ω), i.e. λ < (4 − n)/2,cf. Corollary 1.22(iii) which holds for λ non-integer in place of k, as already men-tioned in Section 1.2.3. Then (3.48) indeed defines F ∈ W 2−λ,2(Ω)∗; note that


F : (μ, η) �→ F : M (Ω) × M (Γ) → W 2−λ,2(Ω)∗ defined by (3.48) is the adjointmapping to v �→ (v, v|Γ) : W 2−λ,2(Ω) → C(Ω) × C(Γ), so we have F = F(μ, η).In this notation, u = B∗F = F ◦ B ∈ Wλ,2(Ω)∗∗ ∼= Wλ,2(Ω) is a solution to〈g, u〉 = 〈F,Bg〉. Moreover, because g is arbitrary, this solution must be unique.Also, we have the estimate

‖u‖Wλ,2(Ω) ≤ K‖F‖W 2−λ,2(Ω)∗ ≤ NK‖(μ, η)‖M (Ω)×M (Γ) (3.49)

with N the norm of the embedding W 2−λ,2(Ω) ⊂ C(Ω). �

Let us realize the embedding Wλ,2(Ω) ⊂ Lq(Ω) with q from (3.44), cf. Corol-lary 1.22(i) for λ non-integer in place of k; more precisely, for any q < n/(n−2)we can choose λ < (4−n)/2. Let us also note that W 1,n/(n−1)+ε(Ω) mentioned inRemark 3.24 is embedded into it, too, i.e. (n/(n−1) + ε)∗ = q < n/(n−2) if ε > 0is taken small.

Lemma 3.30 (Wλ,2-estimate). If c : Ω × R → R satisfies (3.44), g ∈ L2∗′(Ω)

and h ∈ L2#′(Γ), then the conventional weak solution u ∈ W 1,2(Ω) of the equation

−div(A(x)∇u)+c(x, u) = g with the boundary conditions νA(x)∇u+b1(x)u = hsatisfies, for any λ < 2− n/2, also the a-priori estimate

‖u‖Wλ,2(Ω) ≤ C(‖g‖L1(Ω) + ‖h‖L1(Γ) + ‖γ0‖L1(Ω) + ‖γ1‖L1(Ω)

). (3.50)

Proof. Use the test v := signε(u), ε > 0, see (3.23) for the regularized signumfunction. Passing ε → 0, we get εc

∫Ω |c(u)|dx+ b0

∫Γ |u| ≤ ‖g‖L1(Ω) + ‖h‖L1(Γ) +

‖γ1‖L1(Ω); realize that c(x, r)sign(r) ≥ εc|c(x, r)| − γ1(x) and cf. Step 2 in theproof of Proposition 3.22. In particular, ‖g−c(u)‖L1(Ω) ≤ ‖g‖L1(Ω)+‖c(u)‖L1(Ω) ≤(1 + ε−1

c )‖g‖L1(Ω) + ε−1c ‖h‖L1(Γ) + ε−1

c ‖γ1‖L1(Ω). Then one can use Lemma 3.29for μ := g − c(u) and η := h. �

Proposition 3.31 (Existence and stability). Let Ω be a C2-domain, A ∈C1(Ω;Rn×n) satisfy (2.170), b1 ∈ W 1,2#2/(2#−2)(Γ), b1(x) ≥ b0 > 0, and c satisfy(3.44). Then the problem (3.43) has a distributional solution. Moreover, for any

sequences {gk}k∈N ⊂ L2∗′(Ω) and {hk}k∈N ⊂ L2#

′(Γ) converging respectively to the

measures μ and η weakly*, the sequence {uk}k∈N ⊂ W 1,2(Ω) of the correspond-ing weak solutions contains a subsequence converging weakly in Wλ,2(Ω) for anyλ < 2− n/2 to some u and any u obtained by this way is a distributional solutionto (3.43).

Proof. It suffices to select a subsequence converging weakly in Wλ,2(Ω)and to make a limit passage in the integral identity

∫Ω(c(x, uk) − gk)v −

ukdiv(A(x)∇v) dx =∫Γhkv dS, which just gives (3.45); realize the compact

embedding Wλ,2(Ω) ⊂ Ln/(n−2)−ε(Ω) for any ε > 0 and λ < 2 − n/2 largeenough with respect to this ε, and the continuity of the Nemytskiı mappingNc : L

n/(n−2)−ε(Ω) → L1(Ω) provided ε := n/(n−2)− q. �


Remark 3.32 (The case μ and η absolutely continuous). If μ and η are absolutelycontinuous (and g and h are the respective densities), and r �→ c(x, r)− εr nonde-creasing for some ε > 0, then the distributional solution is the “accretive” solution.Moreover, (3.50) yields an additional estimate of u.

3.3 Exercises

Exercise 3.33. Show that u+A(u) = f has a unique solution if A is m-accretive.19

Exercise 3.34. Show what the mapping (u, v) �→ sup⟨u, J(v)

⟩=:⟨u, v

⟩sis upper

semicontinuous with respect to the norm topology on X ×X .20

Exercise 3.35. Prove the formula (3.21), assuming the uniform convexity ofW 1,p(Ω) known.21

Exercise 3.36 (The comparison technique). Prove the estimate u ≤ G for u solv-ing (3.30) in the weak sense by testing v = (u − G)+ with G as suggested inRemark 3.19.22

Exercise 3.37 (Maximal accretivity). Show that m-accretive mappings are maximalaccretive.23

19Hint: Take u1 and u2 two solutions, subtract the corresponding equations, test it by J(u1 −u2), and use (3.8).

20Hint: Take uk → u and vk → v, and j∗k ∈ J(vk) such that 〈j∗k , uk〉 = sup〈J(vk), uk〉 (such j∗kdoes exist because J(vk) is weakly* compact), then take a subsequence j∗k ⇀ j∗ weakly* in X∗(such a subsequence exists because {J(vk)} is bounded), and use Lemma 3.2(i), to show j∗ ∈J(v). Then also sup〈J(vk), uk〉 = 〈j∗k , uk〉 → 〈j∗, u〉 ≤ sup〈J(v), u〉. As this holds for an arbitrarycluster point of {j∗k}, we proved the desired upper semicontinuity lim supk→∞ sup〈J(vk), uk〉 ≤sup〈J(v), u〉.

21Hint: The modification of (3.19) is routine, while (3.20) needs additionally the estimate

⟨|u|p−2u−Δpu, v⟩

=

∫Ω|∇u|p−2∇u · ∇v + |u|p−2uv dx

≤ ‖∇u‖p−1Lp(Ω;Rm)

‖∇v‖Lp(Ω;Rm) + ‖u‖p−1Lp(Ω)

‖v‖Lp(Ω)

≤ (‖∇u‖pLp(Ω;Rm)

+ ‖u‖pLp(Ω)

)p−1(‖∇v‖pLp(Ω;Rm)

+ ‖v‖pLp(Ω)

)1/p= ‖u‖p−1

W1,p(Ω)

∥∥v∥∥W1,p(Ω)

.

22Hint: First, consider rather the modified (but equivalent) equation (u−G) − λdiv a(∇u) +λc(u) = g − G with the boundary condition ν · a(∇u) + b(u) = b(b−1(h)). Then test it byv = (u−G)+ and realize that ∇v = χ{u>G}∇u, cf. (1.50). This yields∫

Ω

((u−G)+

)2+ λc(u)(u−G)+dx+

∫{x∈Ω; u(x)>G}

λa(∇u) · ∇udx

+

∫Γλ(b(u) − b

(b−1(h)

))(u−G)+dS =

∫Ω(g −G)(u−G)+dx ≤ 0.

Note that (b(u) − b(b−1(h)))(u −G)+ ≥ 0 since b(x, ·) is monotone. Also note that p∗ ≥ 2 is tobe used.

23Hint: Take an m-accretive mapping A0, some other accretive mapping A and (u, f) such that

3.3. Exercises 113

Exercise 3.38 (Accretivity of Laplacean in W 1,q). Show the m-accretivity of A =−Δ with dom(A) := {u ∈ W 1,q

0 (Ω); Δu ∈W 1,q0 (Ω)} in X := W 1,q

0 (Ω) if Ω is aC2-domain and if q∗ ≥ 2 and q ≤ 2∗.24

Exercise 3.39 (Renorming of Lp(Ω;Rm) and W 1,p0 (Ω)). Consider an equivalent

norm on Lp(Ω;Rm) given by ‖v‖Lp(Ω;Rm) := (∫Ω

∑mi=1 |vi|pdx)1/p and derive that

‖v‖Lp(Ω;Rm)∗=(∫Ω

∑mi=1 |vi|p

′dx)1/p

′and that the duality mapping J is given by25

[J(u)

]i(x) = ui(x)

∣∣ui(x)∣∣p−2

/∥∥u∥∥p−2

Lp(Ω;Rm), i = 1, . . . ,m. (3.51)

Moreover, consider W 1,p0 (Ω) normed by ‖v‖W 1,p

0 (Ω) := (∫Ω

∑ni=1 |∇vi|pdx)1/p and

derive that J now involves the “anisotropic” p-Laplacean, cf. Example 4.31, namely

J(u) = −div

(∣∣∣ ∂u∂x1

∣∣∣p−2 ∂u

∂x1, . . . ,

∣∣∣ ∂u∂xn

∣∣∣p−2 ∂u

∂xn

)/∥∥u∥∥p−2

W 1,p0 (Ω)

. (3.52)

Exercise 3.40. Derive the very weak formulation (3.45)–(3.46) by applying Green’sformula twice to (3.43).

Exercise 3.41. Considering f ∈ L1(Ω), show that the infimum of the functionalu �→ ∫

Ω12 |∇u|2 − fu dx on W 1,2

0 (Ω) is −∞ if and only if f �∈ L2∗′(Ω).26

Exercise 3.42 (Selectivity for the distributional solution). Show that, if the dataare smooth enough, then any u ∈ C2(Ω) satisfying (3.45)–(3.46) solves (3.43).27

(u, f) ∈ graph(A) ⊃ graph(A0), and take v ∈ dom(A0) such that v + A0(v) = u + f , and thenfrom (3.8) for λ = 1 deduce that ‖u − v‖ ≤ ‖u+ A(u) − v − A(v)‖ = ‖u+ f − v − A0(v)‖ = 0,hencefore u = v and also (u, f) ∈ graph(A0).

24Hint: Taking into account (3.18), show that 〈J(u−v), A(u)−A(v)〉 =∫Ω div(|∇(u −

v)|q−2∇(u−v))Δ(u−v) dx = (q−1)|∇(u−v)|q−2|Δ(u−v)|2dx ≥ 0. Further, show that the weak

solution to u−Δu = g ∈ W 1,q0 (Ω) is in W 1,q

0 (Ω) because, by Proposition 2.104 used for g ∈ L2(Ω)

if q∗ ≥ 2 and modified for zero-Dirichlet boundary condition, it holds u ∈ W 2,2(Ω)∩W 1,20 (Ω) ⊂

W 1,q0 (Ω) if 2∗ ≥ q.25Hint: Derive the corresponding Holder inequality as in (1.19) but now from the Young

inequality of the form∫Ω

∑mi=1 uividx ≤

∫Ω

∑mi=1(

1p|ui|p + 1

p′ |vi|p′) dx. From this, derive the

norm of the dual space. Eventually, modify the calculations in Proposition 3.13.26Hint: Assuming inf

u∈W1,20 (Ω)

∫Ω

12|∇u|2 − fu dx ≥ −C > −∞, realize that

‖f‖W

1,20 (Ω)∗ = sup

‖u‖W

1,20 (Ω)

≤1

∫Ωfu dx ≤ sup

‖u‖W

1,20 (Ω)

≤1

∫Ω

1

2|∇u|2 dx+ C =

1

2+ C < +∞

and therefore necessarily f ∈ L2∗′(Ω).

27Hint: Assume in particular the measures μ and η to have respectively densities g ∈ C(Ω) and



The concept of accretivity for nonlinear mappings was invented essentially byKato [225, 226]; independently Browder [74] invented it in a stronger variant re-quiring (3.6) to hold for any f ∈ J(u−v). Although in the theory of partial dif-ferential equations the accretivity concept is not the dominant one in comparisonwith monotonicity, there are a lot of monographs fully or at least partly devotedto accretive mappings, mainly Barbu [37], Browder [76], Cazenave and Haraux[90, Chap.2-3], Cioranescu [95, Chap.VI], Deimling [118, Chap.13], Hu and Pa-pageorgiou [209, Part I, Sect.III.7], Ito and Kappel [212, Chap.1], Miyadera [287,Chap.2], Pavel [326], Showalter [383, Chap.4], Vainberg [414, Chap.VII], Yosida[425, Sect.XIV.6-7], and Zeidler [427, Chap.31 and 57]. A generalization for set-valued accretive mappings28 exists, too; cf. [118, 209].

The duality mapping was introduced by Beurling and Livingston [51], cf. also[22, 64], and the above listed monographs.

The transposition method exposed in Section 3.2.5 was invented by Stam-pacchia [393], and thoroughly developed especially by Lions and Magenes [262]for linear problems. Semilinear equations with measures in the right-hand sideare investigated by Amann and Quittner [16], Attouch, Bouchitte, and Mambrouk[24], or Brezis [67]. A counterexample of nonuniqueness is due to Serrin [381].

Quasilinear equations with measures in the right-hand side were attackedby Boccardo and Gallouet [56, 57], showing that there is a unique weak solutionu ∈ W 1,q(Ω) with q < n(p−1)/(n−1) with p referring to the growth of the principalpart. In particular, for the problem (3.43) it gives u ∈ W 1,q(Ω) with any q <n/(n−1) which is just embedded into Lp(Ω) with p < n/(n−2) as taken in (3.44).A nonexistence result for c of a growth bigger than (3.44) in case n ≥ 3 is dueto Brezis and Benilan [67] while a counterexample of nonstability is in [24]. Otherdefinitions of solutions have been scrutinized by Boccardo, Gallouet and Orsina[58], Dal Maso, Murat, Orsina, and Prignet [117], and Rakotoson [345]. For thistopic, see also Dolzmann, Hungerbuhler, and Muller [125] or the monograph byMaly and Ziemer [271, Sect.4.4].

h ∈ C(Γ), apply the Green formula twice to the residuum in (3.45), and use (3.46) to obtain:

0 =∫Ω

(c(u)− g

)v − u div

(A∇v

)dx− ∫

Γhv dS

=∫Ω

(c(u)− g − div

(A∇u)

)v dx+

∫Γ

(A∇u·ν − h

)v − u

(A∇v·ν) dS

=∫Ω

(c(u)− g − div

(A∇u)

)v dx+

∫Γ

(A∇u·ν + b1u− h

)v dS.

Then use v ∈ C0(Ω) to get c(u) − div(A∇u)

)= g on Ω, and eventually a general v to get the

boundary condition νA∇u+ b1u = h on Γ.28A set-valued mapping A : dom(A) ⇒ V , dom(A) ⊂ V , is called accretive if ∀u, v ∈ dom(A)

∀u1 ∈ A(u), v1 ∈ A(v) ∃j∗ ∈ J(u − v): 〈j∗, u1 − v1〉 ≥ 0. Moreover, an accretive mappingA : dom(A) ⇒ V , dom(A) ⊂ V , is called m-accretive if I + A is surjective, i.e. ∀f ∈ V ∃v ∈ V :v + A(v) � f . It holds that A is m-accretive if and only if I + λA is surjective for some (or forall) λ > 0.

Chapter 4

Potential problems: smooth case

Again, we consider V a reflexive and separable Banach space. Here we shall dealwith the case that A : V → V ∗ has the form

A = Φ′ (4.1)

for some functional (called a potential) Φ : V → R, having the Gateaux differential1

denoted by Φ′ : V → V ∗. The methods based on the hypothesis (4.1) are calledvariational methods.2

4.1 Abstract theory

Definition 4.1. Let Φ : V → R. Then:

(i) Φ is called coercive if lim‖u‖→∞ Φ(u)/‖u‖ = +∞.

(ii) Φ : V → R is called weakly coercive if lim‖u‖→∞ Φ(u) = +∞.

(iii) u ∈ V is a critical point for Φ if Φ′(u) = 0.

Obviously, solutions to the equation A(u) = f are just the critical points ofthe functional u �→ Φ(u)− 〈f, u〉. The coercive potential problems can be treatedby a so-called direct method based on the Bolzano-Weierstrass Theorem 1.8.

Theorem 4.2 (Direct method). Let Φ : V → R be Gateaux differentiable andweakly lower semicontinuous, and A = Φ′. Then:

1Let us recall that, by definition (1.10), Φ has Gateaux differential at u if the directionalderivative DΦ(u, h) = limε↘0(Φ(u + εh) − Φ(u))/ε does exist for any h ∈ V and DΦ(u, ·) is alinear and continuous functional, denoted just by Φ′(u) ∈ V ∗.

2Sometimes, the notion “variational methods” is used in a wider sense for the setting usingan operator A : V → V ∗, in contrast to non-variational methods as in Chapter 3 where, e.g.,even the direct method may completely fail in general, cf. Exercise 3.41.


116 Chapter 4. Potential problems: smooth case

(i) If Φ is weakly coercive, the equation A(u) = 0 has a solution.

(ii) If Φ is coercive then, for any f ∈ V ∗, the equation A(u) = f has a solution.

(iii) If Φ is strictly convex, then A(u) = f has at most one solution.

Proof. Take a minimizing sequence {uk}k∈N for Φ, i.e.

limk→∞

Φ(uk) = infv∈V

Φ(v); (4.2)

such a sequence does exist by the definition of the infimum3. As Φ is weaklycoercive, {uk}k∈N is bounded. As V is assumed reflexive and separable (hencealso V ∗ is separable, cf. Proposition 1.3), by the Banach Theorem 1.7 it has aweakly convergent subsequence, say uk ⇀ u. As Φ is weakly lower semicontinuous4,Φ(u) ≤ lim infk→∞ Φ(uk) = limk→∞ Φ(uk) = minv∈V Φ(v). Suppose Φ′(u) �= 0,then for some h ∈ V we would have 〈Φ′(u), h〉 = DΦ(u, h) < 0 so that, for asufficiently small ε > 0, we would have

Φ(u+ εh) = Φ(u) + ε⟨Φ′(u), h

⟩+ o(ε) < Φ(u), (4.3)

a contradiction. Thus A(u) = Φ′(u) = 0.If Φ is coercive, then u �→ Φ(u) − 〈f, u〉 is weakly coercive for any f ∈ V ∗.

Yet, this functional obviously has the gradient A− f .If Φ is also convex, then any solution to A(u) = f must minimize Φ − f .

Having two solutions u1 and u2 and supposing u1 �= u2, then, by strict convexity,we get[

Φ−f](1

2u1 +

1

2u2

)<

1

2

[Φ−f

](u1) +

1

2

[Φ−f

](u2) = min

v∈V[Φ−f

](v), (4.4)

a contradiction showing u1 = u2. �Corollary 4.3. Let Φ := Φ1+Φ2 be coercive with Φ1 convex, and Gateaux differen-tiable, and with Φ2 weakly continuous, and Gateaux differentiable. Then, for anyf ∈ V ∗, the equation A(u) = f has a solution.

Proof. Φ1 convex and smooth implies that Φ1 is weakly lower semicontinuous:indeed, by convexity always Φ1(u) + 〈Φ′1(u), v − u〉 ≤ Φ1(v), cf. (4.12) below, sothat

Φ1(u) ≤ lim infv⇀u

(Φ1(v) + 〈Φ′1(u), u− v〉) = lim inf

v⇀uΦ1(v) (4.5)

because limv⇀u〈Φ′1(u), u− v〉 = 0. Thus Φ1 +Φ2 is weakly lower semicontinuous.Then one can use the previous Theorem 4.2(ii). �

3Note also that infv∈V Φ(v) > −∞ otherwise, by weak coercivity and weak lower semiconti-nuity of Φ, there would exist v such that Φ(v) = −∞, which contradicts Φ : V → R.

4Recall our convention that by (semi)continuity, see (1.6), we mean what is sometimes called“sequential” (semi)continuity while the general concept of (semi)continuity works with general-ized sequences (nets). For our purposes, the sequential concept is relevant. Additionally, as Φis coercive and V separable, both modes of lower semicontinuity of Φ coincide with each otherbecause the weak topology on bounded sets is metrizable.


Theorem 4.4 (Relations between A and Φ). Let Φ : V → R be Gateauxdifferentiable, and A = Φ′. Then:

(i) If A is coercive and monotone, then also Φ is coercive.

(ii) If A is pseudomonotone, then Φ is weakly lower semicontinuous.

(iii) If A is (strictly) monotone, then Φ is (strictly) convex, weakly lower semicon-tinuous and locally Lipschitz continuous.

(iv) Conversely, if Φ is convex (resp. strictly convex), then A is monotone (resp.strictly monotone).

Proof. The point (i): First, let us realize that5

Φ(u) = Φ(0) +

∫ 1

0

⟨A(tu), u

⟩dt (4.6)

because, denoting ϕ(t) = Φ(tu), one has

Φ(u)− Φ(0) = ϕ(1)− ϕ(0) =

∫ 1

0

ϕ′(t) dt =∫ 1

0

limε→0

Φ(tu+εu)− Φ(tu)

εdt

=

∫ 1

0

DΦ(tu, u) dt =

∫ 1

0

〈Φ′(tu), u〉dt =∫ 1

0

〈A(tu), u〉dt. (4.7)

Then, supposing for simplicity Φ(0) = 0, one gets6

Φ(u) =

∫ 1

0

⟨A(tu), u

⟩dt =

∫ 1

0

〈A(tu)−A(0), tu− 0〉t

dt+ 〈A(0), u〉

≥∫ 1

1/2

(〈A(tu), tu〉t

− 〈A(0), u〉)dt+ 〈A(0), u〉

=

∫ 1

1/2

〈A(tu), tu〉t

dt+〈A(0), u〉

2≥∫ 1

1/2

ζ(‖tu‖)‖tu‖t

dt+〈A(0), u〉

2

≥∫ 1

1/2

ζ(‖ 12u‖)‖tu‖

tdt+

〈A(0), u〉2

≥ 1

2

(ζ(‖ 1

2u‖)− ‖A(0)‖∗

)‖u‖,

where ζ(·) is a nondecreasing function with lims→+∞ ζ(s) = +∞ from Definition2.5. Thus we proved a super-linear growth of Φ.

The point (ii): Suppose the contrary, i.e. Φ is not weakly lower semicontinuousat some point, say at 0; i.e. there are some δ > 0 and a sequence {uk}k∈N ⊂ V ,uk ⇀ 0, such that:

∀k∈N : δ ≤ Φ(0)− Φ(uk). (4.8)

5Note that, as ϕ : t �→ Φ(tu) is convex and finite, hence ϕ′ : t �→ 〈A(tu), u〉, being nonde-creasing, is a Borel function, hence measurable. The integral is finite as Φ is finite, cf. (4.7).

6Note that∫ 10 t−1〈A(tu)−A(0), tu−0〉 dt ≥ ∫ 1

1/2 t−1〈A(tu)−A(0), tu−0〉 dt as A is monotone.


For v, u ∈ V , put ϕ(t) = Φ(u+ tv). By the mean value theorem, there is t ∈ (0, 1)such that ϕ(1)− ϕ(0) = ϕ′(t), i.e.

Φ(v + u)− Φ(u) =⟨Φ′(u + tv), v

⟩. (4.9)

Take ε > 0. By (4.8) and using (4.9) with v := εuk and u := 0, we have

δ ≤ Φ(0)− Φ(εuk) + Φ(εuk)− Φ(uk) = Φ(εuk)− Φ(uk)− ε⟨Φ′(εtk,εuk), uk

⟩,

where tk,ε ∈ (0, 1) depends on ε and on k. As {uk}k∈N, being weakly convergent,is bounded7 and Φ′, being pseudomonotone, is bounded on bounded subsets, thelast term is O(ε). Consider ε > 0 fixed and so small that this last term is less thanδ/2 for all k ∈ N, hence, for a suitable tk ∈ (0, 1), one has

δ

2≤ Φ(εuk)− Φ(uk) =

⟨Φ′(wk), (ε−1)uk

⟩=

(ε−1)⟨Φ′(wk), wk

⟩1− tk(1−ε)

(4.10)

where we abbreviated wk := uk − tk(1−ε)uk and where (4.9) was used for v :=(ε− 1)uk and u := uk. From (4.10), we have

lim supk→∞

⟨Φ′(wk), wk

⟩= lim sup

k→∞

1− tk(1−ε)

1− ε

(Φ(uk)− Φ(εuk)

) ≤ 1

1−ε

(−δ

2

)< 0.

As uk ⇀ 0, we have also wk ⇀ 0. By using the pseudomonotonicity (2.3b) forA = Φ′, v := 0, and the sequence {wk}k∈N, we have lim infk→∞〈Φ′(wk), wk〉 ≥〈Φ′(0), 0〉 = 0, which is the sought contradiction.

The point (iii): Monotonicity of A implies its local boundedness, seeLemma 2.15, and thus Φ, having a locally bounded derivative, is locally Lips-chitz continuous. Denote ϕw(t) := Φ(u+tw). As A is assumed monotone, ϕ′w(t) =〈A(u+tw), w〉 is nondecreasing because obviously, for t2 > t1,

ϕ′w(t2)− ϕ′w(t1) =⟨A(u+t2w)−A(u+t1w), w

⟩=

⟨A(u+t2w)−A(u+t1w), (u+t2w) − (u+t1w)

⟩t2 − t1

≥ 0. (4.11)

Then we have

Φ(v)− Φ(u) = ϕv−u(1)− ϕv−u(0) =

∫ 1

0

ϕ′v−u(t) dt

≥∫ 1

0

ϕ′v−u(0) dt = ϕ′v−u(0) =⟨A(u), v − u

⟩. (4.12)

Put z := 12u+ 1

2v. By (4.12) we have

Φ(u) ≥ Φ(z) +⟨A(z), u− z

⟩and Φ(v) ≥ Φ(z) +

⟨A(z), v − z

⟩. (4.13)

7Here we use the Banach-Steinhaus principle, see Theorem 1.1.


Adding these inequalities and recalling the lower semicontinuity of Φ (cf. Exer-cise 4.18), one gets the convexity of Φ:

Φ(u) + Φ(v) ≥ 2Φ(z) +⟨A(z), u−z

⟩+⟨A(z), v−z

⟩= 2Φ(z). (4.14)

Strict monotonicity of A implies that ϕw is increasing for w �= 0, and then (4.12)and (4.13) hold with strict inequalities provided v �= u. Eventually, from (4.12),the weak lower semicontinuity of Φ has already been proved in (4.5).

The point (iv): Φ convex implies ϕw : t �→ Φ(u + tw) convex, and thereforeϕ′w(t) = 〈A(u + tw), w〉 is nondecreasing. Thus, for w = v − u, we get⟨

A(v) −A(u), v−u⟩= ϕ′v−u(1)− ϕ′v−u(0) ≥ 0, (4.15)

so that A is monotone. If Φ is strictly convex, so is ϕw if w �= 0, and thus ϕ′w isincreasing, hence A strictly monotone. �

Remark 4.5. The coercivity of Φ does not imply coercivity of A.8

Corollary 4.6. Let A = Φ′, and one of the following two sets of conditions holds:

(i) A is pseudomonotone and Φ is coercive, or

(ii) A is monotone and coercive.

Then the equation A(u) = f has a solution for any f ∈ V ∗.

Proof. As to (i), Theorem 4.4(ii) implies the weak lower semicontinuity of thecoercive potential Φ of A. Then use Theorem 4.2(ii).

As to (ii), Theorem 4.4(i) and (iii) implies the coercivity and the weak lowersemicontinuity of the potential Φ of A. Then again use Theorem 4.2(ii). �

Remark 4.7. Comparing Corollary 4.6 with Theorem 2.6, we can see that now werequired in addition the potentiality, but got a more constructive proof avoidingthe Brouwer fixed-point Theorem 1.10. Similarly, comparison with Browder-MintyTheorem 2.18 yields that potentiality makes no need for any explicit assumptionof radial continuity9 of A.

Remark 4.8 (Potentiality criteria). If A is hemicontinuous and Gateaux differ-entiable, it has a potential (given then by the formula (4.6)) if and only if〈[A′(u)](v), w〉 = 〈[A′(u)](w), v〉 for any u, v, w ∈ V . In the general case, the fol-lowing integral criterion is sufficient and necessary for potentiality of A:∫ 1

0

⟨A(tu), u

⟩dt−

∫ 1

0

⟨A(tv), v

⟩dt =

∫ 1

0

⟨A(v + t(u−v)), u−v

⟩dt. (4.16)

8Indeed, any Φ coercive can be changed in a neighbourhood of {nu; u �= 0, n ∈ N} to belocally constant; then A(nu) = 0 so that A will not be coercive while Φ remains coercive.

9In fact, this is just an “optical” illusion as every monotone and potential mapping is evendemicontinuous, cf. e.g. Gajewski at al. [168, Ch.III,Lemma 4.12].


Remark 4.9 (Iterative methods). The existence of a potential suggests iterativemethods for minimization of Φ to solve the equation A(u) = f . E.g., if V isuniformly convex and V ∗ strictly convex, the abstract steepest-descent-like methodlooks as

uk+1 := uk + εkJ−1(f − Φ′(uk)

)(4.17)

with εk > 0 sufficiently small; e.g. εk := min (1, 2/(ε+ ‖uk‖+ ‖Auk−f‖∗)) for the Lipschitz constant of A and ε > 0 guarantees global strong convergence incase A is strictly monotone and d-monotone, Lipschitz continuous, and coercive.10

Note that, if V ≡ V ∗ is a Hilbert space, J−1(f − Φ′(uk)) = f − Φ′(uk) is justthe steepest descent of the landscape given by the graph of Φ− f , which gave thename to this method; cf. also (4.17) with (2.43) or with Example 2.93.

Remark 4.10 (Ritz’ method [350]). Assuming {Vk}k∈N is a nondecreasing sequenceof finite-dimensional subspaces of V whose union is dense, see (2.7), we can considera sequence of problems

Find uk ∈ Vk : Φ(uk) = minv∈Vk

Φ(v) . (4.18)

Note that uk is simultaneously a Galerkin approximation to the equation A(u) = 0with A = Φ′, see (2.8). The Ritz method can be combined with (4.17) to get acomputer implementable strategy, although much more efficient algorithms than(4.17) are usually implemented.

Remark 4.11 (Quadratic case). A very special case is that Φ is quadratic:

Φ(u) =⟨12Au− f, u

⟩(4.19)

with A ∈ L (V, V ∗), A∗ = A. Then A is weakly continuous, so the existence ofa solution to Au = f follows simply by Section 2.5 if A (or, equivalently, Φ) iscoercive, which here reduces to positive definiteness of A, i.e. 〈Av, v〉 ≥ ε‖v‖2for some ε > 0 and all v ∈ V , cf. also the Lax-Milgram Theorem 2.19 wherehowever A∗ = A was redundant. Here, alternatively, Au = f is just equivalentwith Φ′(u) = 0 and a direct method can be applied, too.

4.2 Application to boundary-value problems

The method of mappings possessing a potential has powerful applications inboundary-value problems. However, the requirement (4.1) brings a certain restric-tion on problems that can be treated in this way. Considering again the boundary-value problem (2.49) for the quasilinear 2nd-order equation (2.45), i.e.

−div(a(x, u,∇u)

)+ c(x, u,∇u) = g in Ω,

u|Γ = uD on ΓD,ν · a(x, u,∇u) + b(x, u) = h on ΓN,

⎫⎬⎭ (4.20)

10See Gajewski at al. [168, Thm.III.4.2] using the (S)-property which is here implied by d-monotonicity with uniform convexity of V , cf. Remark 2.21.

4.2. Application to boundary-value problems 121

we will assume ai(x, ·, ·), i = 1, . . . , n, and c(x, ·, ·) smooth in the sense W 1,1loc (R×

Rn), and impose the symmetry conditions

∂ai(x, r, s)

∂sj=

∂aj(x, r, s)

∂si,

∂ai(x, r, s)

∂r=

∂c(x, r, s)

∂si(4.21)

for all 1 ≤ i ≤ n, 1 ≤ j ≤ n, and for a.a. (x, r, s) ∈ Ω × R × Rn. In other words,(4.21) says that the Jacobian matrix of the mapping

(r, s) �→ (c(x, r, s), a(x, r, s)

): R1+n → R1+n (4.22)

is symmetric for a.a. x ∈ Ω. Let us emphasize that (4.21) is not necessary for Ato have a potential11. Yet, we will prove that, if (4.21) holds, then the mapping Adefined by (2.59) has a potential Φ : W 1,p(Ω) → R in the form

Φ(u) =

∫Ω

ϕ(x, u,∇u) dx +

∫ΓN

ψ(x, u) dS , where (4.23a)

ϕ(x, r, s) =

∫ 1

0

s·a(x, tr, ts) + r c(x, tr, ts) dt , (4.23b)

ψ(x, r) =

∫ 1

0

r b(x, tr) dt ; (4.23c)

for the derivation of (4.23b,c) from the formula (4.6), cf. Exercise 4.22. In thissituation, (4.20) is called the Euler-Lagrange equation for (4.23).

Lemma 4.12 (Continuity of Φ). Let (2.55) with ε = 0 hold. Then Φ is contin-uous.

Proof. The particular terms of Φ are continuous as a consequence of the continuityof the mapping u �→ ∇u : W 1,p(Ω) → Lp(Ω;Rn), of the embedding W 1,p(Ω) into

Lp∗(Ω) and of the trace operator u �→ u|Γ : W 1,p(Ω) → Lp#

(ΓN) provided thecontinuity of the Nemytskiı operators Nϕ : Lp∗

(Ω) × Lp(Ω;Rn) → L1(Ω) and

Nψ : Lp#

(ΓN) → L1(ΓN) would be ensured. As to Nϕ, the needed growth conditionon ϕ looks as

∃γ ∈ L1(Ω) ∃C ∈ R : |ϕ(x, r, s)| ≤ γ(x) + C|r|p∗+ C|s|p. (4.24)

In view of (4.23b), the condition (4.24) is indeed ensured by (2.55a,c) even weak-

11E.g. if n = 1 and a = a(x, s) and c = c(x, r), then the basic Caratheodory hypothesis isobviously sufficient; ϕ(x, ·, ·) is just the sum of the primitive functions of a(x, ·) and c(x, ·). Ingeneral, (4.21) holding only in the sense of distributions suffices, see Necas [305, Theorem 3.2.12].


ened by putting ε = 0 because of the estimate

|ϕ(x, r, s)| ≤∫ 1

0

|s · a(x, tr, ts)|dt+∫ 1

0

|rc(x, tr, ts)|dt

≤∫ 1

0

(|s|(γa(x) + C|tr|p∗/p′

+ C|ts|p−1)

+ |r|(γc(x) + C|tr|p∗−1 + C|ts|p/p∗′))dt

≤ 2|s|pp

+γa(x)

p′

p′+

Cp′ |r|p∗

p′(p∗ + 1)+

C|s|pp

+2|r|p∗

p∗+

γc(x)p∗′

p∗′+

C|r|p∗

p∗+

Cp∗′ |s|pp∗′(p+ 1)

, (4.25)

which obviously requires γa ∈ Lp′(Ω) and γc ∈ Lp∗′

(Ω) as indeed used in (2.55a)and (2.55c), respectively. Then (4.24) obviously follows.

The growth conditions for ψ, i.e. |ψ(x, r)| ≤ γ(x) + C|r|p#

with some γ ∈L1(Γ), can be treated analogously, resulting in (2.55b) with ε = 0. �

Lemma 4.13 (Differentiability of Φ). Let (2.55) for ε = 0 and (4.21) hold,let a(x, ·, ·) ∈ W 1,1

loc (R×Rn;Rn) and c(x, ·, ·) ∈ W 1,1loc (R×Rn) for a.a. x ∈ Ω. Then

Φ is Gateaux differentiable and Φ′ = A with A given by (2.59).

Proof. The directional derivative DΦ(u, v) of Φ at u in the direction v is

DΦ(u, v) := limε→0

Φ(u+ εv)− Φ(u)

ε=

d

dεΦ(u + εv)

∣∣∣ε=0

=d

dε

(∫Ω

ϕ(x, u + εv,∇u+ ε∇v) dx+

∫ΓN

ψ(x, u + εv) dS

)∣∣∣∣ε=0

=

∫Ω

n∑i=1

∂ϕ(x, u,∇u)

∂si

∂v

∂xi+

∂ϕ(x, u,∇u)

∂rv dx+

∫ΓN

∂ψ(x, u)

∂rv dS,

(4.26)

where we have changed the order of integration and differentiation by Theo-rem 1.29. This requires existence of a common (with respect to ε) integrable ma-jorant of the collections

{ϕ′s(u+ εv,∇u+ ε∇v) ·∇v

}0<ε≤ε0

and{ϕ′r(u+ εv,∇u+

ε∇v)v}0<ε≤ε0

and{ψ′r(u + εv)v

}0<ε≤ε0

for some ε0 > 0, where we abbreviated

∂ϕ/∂si =: ϕ′si etc. Assume, for a moment, that

∃γ ∈ Lp′(Ω) ∃C∈R+ : |ϕ′s(x, r, s)| ≤ γ(x) + C|r|p∗/p′

+ C|s|p−1, (4.27a)

∃γ∈Lp#′(ΓN) ∃C∈R+ : |ψ′r(x, r)| ≤ γ(x) + C|r|p#−1, (4.27b)

∃γ∈Lp∗′(Ω) ∃C∈R+ : |ϕ′r(x, r, s)| ≤ γ(x) + C|r|p∗−1 + C|s|p/p∗′

. (4.27c)


As for the first collection, for any ε ∈ [0, ε0] and a suitable Cp depending on p andC from by (4.27a), we have the estimate∣∣ϕ′s(u+εv,∇u+ε∇v) · ∇v

∣∣ ≤ (γ(x) + C|u+εv|p∗/p′+ C|∇u+ε∇v|p−1

)|∇v|≤ (γ(x) + Cp|u|p∗/p′

+ εp∗/p′

0 Cp|v|p∗/p′+ Cp|∇u|p−1+ Cpε

p−10 |∇v|p−1

)|∇v|

which is the sought integrable majorant. The other two terms can be handledanalogously, exploiting respectively (4.27b,c).

Moreover, DΦ(u, ·) : W 1,p(Ω) → R is obviously linear and, by (4.27), alsocontinuous. Hence Φ has the Gateaux differential.

The required form of the Gateaux differential follows from the identities

∂ϕ(x, r, s)

∂si=

∫ 1

0

ai(x, tr, ts) + t

( n∑j=1

sj∂aj∂si

(x, tr, ts) + r∂c

∂si(x, tr, ts)

)dt

=

∫ 1

0

ai(x, tr, ts) + t

( n∑j=1

sj∂ai∂sj

(x, tr, ts) + r∂ai∂r

(x, tr, ts)

)dt

=

∫ 1

0

d

dt

(t ai(x, tr, ts)

)dt =

[t ai(x, tr, ts)

]1t=0

= ai(x, r, s), (4.28)

where (4.23b) with (4.21) has been used; note that by Theorem 1.29 the

change of the order of integration∫ 1

0dt and differentiation ∂

∂siin (4.28) re-

quires a common integrable majorant of {|t �→ ∂∂si

[s ·a(x, tr, ts)]|}|s|≤M and of

{|t �→ ∂∂si

[r c(x, tr, ts)]|}|s|≤M in L1(0, 1) for any M ∈ R, which holds because

a(x, r, ·) ∈ W 1,1loc (R

n;Rn) and c(x, r, ·) ∈ W 1,1loc (R

n) is assumed. Similarly also

∂ϕ(x, r, s)

∂r=

∫ 1

0

c(x, tr, ts) + t

(r∂c

∂r(x, tr, ts) +

n∑i=1

si∂ai∂r

(x, tr, ts)

)dt

=

∫ 1

0

c(x, tr, ts) + t

(r∂c

∂r(x, tr, ts) +

n∑i=1

si∂c

∂si(x, tr, ts)

)dt

=

∫ 1

0

d

dt

(t c(x, tr, ts)

)dt =

[t c(x, tr, ts)

]1t=0

= c(x, r, s). (4.29)

The fact that ∂ψ(x, r)/∂r = b(x, r) can be derived by the easier way, realizingthat (4.23c) defines, in fact, the primitive function of b(x, ·), cf. (4.6). Note that(4.27a) then coincides with the former condition (2.55a), while (4.27b), (4.27c) is(2.55b,c) but weakened with ε = 0, as indeed assumed. �

In the following lemma, we will distinguish whether lower-order terms havecritical growth (and then their monotonicity helps) or whether their growth issub-critical (the cases (i) and (ii) in the following lemma).


Lemma 4.14 (Weak lower semicontinuity of Φ). Let the assumptions ofLemma 4.13 and one of the following conditions be valid:

(i) (2.55) holds for ε > 0 and the assumptions of Lemma 2.32 hold,

(ii) (2.55) holds for ε > 0 and a(x, r, ·) : Rn → Rn is monotone,

(iii) (2.55) holds with ε=0 and the mappings (4.22) and b(x, ·):R→R are monotone.

Then Φ is weakly lower semicontinuous.

Proof. As to the case (i), by Lemmas 2.31–2.32, the gradient A of Φ is pseu-domonotone; here we used also Lemma 4.13. By Theorem 4.4(ii), Φ is weaklylower semicontinuous.

The case (ii): by (4.28), monotonicity of a(x, r, ·) is just monotonicity ofϕ′s(x, r, ·) from which convexity of ϕ(x, r, ·) follows as in the proof of Theo-rem 4.4(iii). Likewise in (4.25), ϕ satisfies the growth condition (4.24) but nowwith p∗ − ε instead of p∗. Hence, considering uk ⇀ u in W 1,p(Ω), by the compactembedding W 1,p(Ω) � Lp∗−ε(Ω), ϕ(uk,∇u) → ϕ(u,∇u) in L1(Ω). Similarly, by(4.28) and (2.55a) with ε > 0, we have ϕ′s(uk,∇u) → ϕ′s(u,∇u) in Lp′

(Ω;Rn).Altogether, by the convexity of ϕ(x, r, ·),

lim infk→∞

∫Ω

ϕ(x, uk,∇uk) dx ≥ limk→∞

∫Ω

ϕ(x, uk,∇u) dx

+ limk→∞

∫Ω

ϕ′s(x, uk,∇u)·(∇uk −∇u) dx =

∫Ω

ϕ(x, u,∇u) dx. (4.30)

Moreover, by compactness of the trace operator u �→ u|Γ : W 1,p(Ω) → Lp#−ε(Γ)

and by the continuity of the Nemytskiı mapping Nψ : Lp#−ε(ΓN) → L1(ΓN),we get the weak continuity of the boundary term in (4.23); the growth of ψ,

i.e. |ψ(x, r)| ≤ γ(x) + C|r|p#−ε, can be estimated as in (4.25).

As to the case (iii), monotonicity of [c, a](x, ·, ·) : R1+n → R1+n impliesconvexity of ϕ(x, ·, ·) : R1+n → R, which can be seen similarly as in the proof ofTheorem 4.4(iii). By monotonicity of b(x, ·), the overall functional Φ is convex. ByLemma 4.13, Φ is smooth so, by (4.5), also weakly lower semicontinuous. �

Lemma 4.15 (Coercivity of Φ). Let us assume measn−1(ΓD) > 0 and 12

a(x, r, s)·s+ c(x, r, s)r ≥ ε1|s|p + ε2|r|q − k0(x)|s| − k1(x)|r|, (4.31a)

b(x, r)r ≥ −k2(x)|r| (4.31b)

with some ε0, ε1 > 0, p ≥ q > 1, and k0 ∈ Lp′(Ω), k1 ∈ Lp∗′

(Ω), and k2 ∈Lp#′

(Γ). Then Φ is coercive on W 1,p(Ω). Besides, Φ is coercive on V = {v ∈W 1,p(Ω); v|ΓD = 0} even if q = 0.

12Cf. (4.31) with (2.92). Note that if one assumes, e.g. b(x, r)r ≥ −k2(x), one would have∫ 10 k2(x)/t dt which is not finite, however.


Proof. In view of (4.23b), one has

ϕ(x, r, s) =

∫ 1

0

s·a(x, tr, ts) + r c(x, tr, ts) dt =

∫ 1

0

ts·a(x, tr, ts)+tr c(x, tr, ts)

tdt

≥∫ 1

0

ε1|ts|p + ε2|tr|q − k0|ts| − k1|tr|t

dt =ε1p|s|p+ ε2

q|r|q− k0|s| − k1|r|.

Similarly, (4.31b) with (4.23c) implies ψ(x, r) ≥ −k2|r|. Then

Φ(u) ≥∫Ω

(ε1p|∇u|p + ε2

q|u|q − k0|∇u| − k1|u|

)dx−

∫Γ

k2|u|dS≥ ε‖u‖qW 1,p(Ω) − C − ‖k0‖Lp′(Ω)‖∇u‖Lp(Ω;Rn)

−‖k1‖Lp∗′(Ω)‖u‖Lp∗(Ω) − ‖k2‖Lp#′(Γ)

‖u‖Lp#(Γ)

≥ ε‖u‖qW 1,p(Ω)−C−(‖k0‖Lp′(Ω)+N1‖k1‖Lp∗′(Ω)+N2‖k2‖Lp#

′(Γ)

)‖u‖W 1,p(Ω)

with ε and C depending on p, q, ε1, ε2 and CP

from the Poincare inequality(1.55), where N1 and N2 stand here respectively for the norms of the embed-

ding W 1,p(Ω) ⊂ Lp∗(Ω) and of the trace operator u �→ u|Γ : W 1,p(Ω) → Lp#

(Γ).As q > 1, the functional Φ is coercive in the sense that Φ(u)/‖u‖W 1,p(Ω) → +∞for ‖u‖W 1,p(Ω) → +∞.

For q = 0, we get coercivity on V by using Poincare’s inequality (1.57). �Proposition 4.16 (Direct method for boundary-value problem (4.20)).Let (4.21), the assumptions of Lemmas 4.13–4.15 hold, and f be defined by(2.60), i.e. 〈f, v〉 :=

∫Ωgv dx +

∫ΓN

hv dS. Then Φ − f has a minimizer on

{v ∈ W 1,p(Ω); v|ΓD = uD}, and every such minimizer solves the boundary-valueproblem (4.20) in the weak sense.

Proof. Let us first transform the problem on the linear space V := {v ∈W 1,p(Ω); v|ΓD = 0}, cf. (2.52), as we did in Proposition 2.27: define Φ0 : v �→Φ(v+w) with w ∈ W 1,p(Ω) such that w|ΓD = uD and f ∈ V ∗ again by (2.60). De-noting A0 := Φ′0, we have A0 : V → V ∗ and A0(v) = A(v +w) with A := Φ′. Thereflexivity13 of W 1,p(Ω) ensures also reflexivity of its closed14 subspace V . Theweak lower semicontinuity and coercivity of Φ, proved respectively in Lemma 4.14and 4.15, is inherited by Φ0, and therefore the existence of a minimizer u0 of Φ0

on V follows by the compactness argument. Then, in view of (4.26), DΦ(u0, v) = 0for any v ∈ V just means that u0 ∈ V solves A0(u0) = f . Then u := u0 +w solvesA(u) = f , i.e. it is the sought weak solution to the boundary-value problem (4.20)cf. Proposition 2.27.

Observing that V + w = {v ∈ W 1,p(Ω); v|ΓD = uD}, one can see that uminimizes Φ− f on V + w. �

13Recall that 1 < p < +∞ is supposed.14Closedness of V follows from the continuity of the trace operator u �→ u|ΓD .


Remark 4.17. In contrast to the non-potential case, Lemma 4.14(ii) allows us totreat nonlinearities of the type c(∇u) without requiring strict monotonicity (2.68a)of a(x, r, ·). Of course, the price for it is the severe restriction (4.21).


Exercise 4.18. Show that lower semicontinuous function f : X → R ∪ {+∞}satisfying 1

2f(u1) +12f(u2) ≥ f(12u1 +

12u2) is convex.

15

Example 4.19 (Duality mapping16). If V ∗ is strictly convex, the duality mappingJ : V → V ∗ has a potential Φ(u) = 1

2‖u‖2. Indeed, we have

⟨J(v), v − u

⟩ ≥ ‖v‖2 − ‖u‖ ‖v‖ ≥ ‖v‖2 − 1

2

(‖u‖2 + ‖v‖2)=

1

2‖v‖2 − 1

2‖u‖2 ≥ ‖u‖ ‖v‖ − ‖u‖2 ≥ ⟨J(u), v − u

⟩. (4.32)

Then put v = u + th. We get 〈J(u + th), th〉 ≥ 12‖u+ th‖2 − 1

2‖u‖2 ≥ 〈J(u), th〉.Divide it by t �= 0, then let t → 0. By the radial continuity17 of J , we come to

⟨J(u), h

⟩= lim

t→0

1

t

(12‖u+ th‖2 − 1

2‖u‖2

)=: DΦ(u, h). (4.33)

Exercise 4.20. Consider (4.21) and the situation in Exercise 2.89 with (2.153), i.e. −b < min(1, εc)/N

2. Show that Φ is strictly convex.

Exercise 4.21 (Abstract Ritz approximation). Consider the Ritz approximation(4.18) and show that uk ⇀ u (for a subsequence) where u solves A(u) = 0,A = Φ′ and, in addition, minimizes Φ over V . Besides, show that Φ(uk) → Φ(u).18

Moreover, assuming Φ = Φ1 + Φ0 with Φ0 weakly continuous and Φ1 such thatΦ1(uk) → Φ1(u) and uk ⇀ u imply uk → u, show that uk → u.19

15Hint: By iterating, show that λf(u1)+(1−λ)f(u2) ≥ f(λu1+(1−λ)u2) not only for λ = 1/2but also for λ = 1/4 and 3/4, and then for any dyadic number in [0, 1], i.e. λ = k2−l, l ∈ N,k = 0, 1, . . . , 2l. Such numbers are dense in [0, 1], and the general case λ ∈ [0, 1] then uses thelower semicontinuity of f .

16The observation that J has a potential is due to Asplund [22].17Recall that we proved even the (norm,weak*)-continuity of J if V ∗ is strictly convex, see

Lemma 3.2(ii).18Hint: For any v∈V take vk∈Vk such that vk → v. Then Φ(uk) ≤ Φ(vk) and, by weak lower

semicontinuity and strong continuity of Φ, it holds that

Φ(u) ≤ lim infk→∞

Φ(uk) ≤ lim infk→∞

Φ(vk) = limk→∞

Φ(vk) = Φ(v).

For a special case v := u, it follows that Φ(uk)→ Φ(u).19Hint: From Φ(uk)→ Φ(u) (already proved) and Φ0(uk)→ Φ0(u) (which follows from weak

continuity of Φ0), deduce Φ1(uk)→ Φ1(u).


Exercise 4.22. Derive (4.23) from (4.6), assuming existence of a potential andusing Fubini’s theorem 1.19.20

Example 4.23. (p-Laplacean.) The operator A(u):=−div(|∇u|p−2∇u

)on W 1,p

0 (Ω)has the potential

Φ(u) =1

p

∫Ω

|∇u(x)|p dx. (4.34)

It just suffices to evaluate (4.23b) with ϕ = ϕ(s):

ϕ(s)=

∫ 1

0

s·a(x, ts) dt =∫ 1

0

s·|ts|p−2ts dt =

∫ 1

0

|s|ptp−1dt = |s|p[tp

p

]1t=0

=|s|pp

.

Exercise 4.24. Verify (4.21) for ai(x, s) := |s|p−2si.21 Show that a ∈ W 1,1

loc (Rn).22

Example 4.25 (More general potentials23). Consider a coefficient σ : R+ → R+

depending on the magnitude of ∇u and the quasilinear mapping

u �→ −div(σ(|∇u|2)∇u

). (4.35)

In application, the concrete form of the function σ(·) > 0 may reflect some phe-nomenology resulting from experiments. Obviously, it fits with our concept forai(x, r, s) := σ(|s|2)si and c ≡ 0. The symmetry condition (4.21) is satisfied and

ϕ(x, r, s) ≡ ϕ(s) =1

2

∫ |s|2

0

σ(ξ) dξ. (4.36)

The monotonicity of the mapping (4.35) is related to positive definiteness of thesecond derivative ϕ′′(s), i.e. ϕ′′(s; s, s) = 2σ′(|s|2)(s · s)2 + σ(|s|2)|s|2 ≥ 0 for anys, s ∈ Rn. This is trivially true if σ′(|s|2) ≥ 0. When estimating (s · s)2 ≥ −|s|2|s|2,one can see that this condition is certainly satisfied if

∀ξ ≥ 0 : σ(ξ) ≥ 2ξmax(−σ′(ξ), 0). (4.37)

Hence σ may increase arbitrarily but must have a limited decay.

20Hint:∫ 10 〈A(tu), u〉dt =

∫ 10 (

∫Ω a(tu, t∇u)·∇u+c(tu, t∇u)udx+

∫ΓN

b(tu)udS) dt=∫Ω

(∫ 10 a

(tu, t∇u)·∇udt+∫ 10 c(tu, t∇u)u dt

)dx+

∫ΓN

∫ 10 b(tu)u dtdS =

∫Ω ϕ(u,∇u) dx+

∫ΓN

ψ(u) dS.21Hint: The symmetry of the matrix a′s(x, s) follows by the direct calculations:

∂ai(x, s)

∂sj=

∂|s|p−2si

∂sj= si(p− 2)|s|p−4sj + |s|p−2δij .

22Hint: Indeed, si(p − 2)|s|p−4sj = O(|s|p−2) for s → 0. Hence this term is integrable alsoaround the origin if p > 1, as assumed.

23Cf. also Malek et al. [268, p.15] or Zeidler [427, Vol.II/B, Lemma 25.26].


Exercise 4.26 (Regularizations of p-Laplacean). Having in mind (4.36) with thecoefficient σ in the analytical form

σ(ξ) := ε1 +(ε2 + ξ

)(p−2)/2or (4.38a)

σ(ξ) := ε1 +(ε2 +

√ξ)p−2

, ε1, ε2 ≥ 0, (4.38b)

show that ϕ′′(s; s, s) ≥ 0 so that (4.35) creates a monotone potential mapping.24

One obviously gets the p-Laplacean when putting ε1 = ε2 = 0 in (4.38) whileε2 > 0 makes its regularization around 0 as shown on Figure 9. The effect of ε1 > 0is just a vertical shift of σ and has already been considered in Exercise 2.89.

(a)

(a)

(b)(b)

(c)

(c)

p = 12/5

p = 7/5

00

00

1 1

2 2

33 66

σ σ

ξ = |∇u|2ξ = |∇u|2

Figure 9. Various dependence of the coefficient σ as a function of |∇u|2;(a)= the case (4.38a) with ε1 = 0 and ε2 = 2,(b)= the case (4.38b) with ε1 = 0 and ε2 = 2,(c)= the case (4.38) with ε1 = ε2 = 0, i.e. the p-Laplacean.

Exercise 4.27 (Convergence of the finite-element method). Consider the boundary-value problem (2.147). Show that it has the potential

Φ(u) =

∫Ω

( |∇u(x)|pp

+

∫ u(x)

0

c(x, r) dr

)dx+

∫Γ

(∫ u(x)

0

b(x, r) dr

)dS (4.39)

and note that no smoothness of b(x, ·) and c(x, ·) is required for (4.39). Assume Ωpolyhedral, take a finite-dimensional Vk as in Example 2.67, and consider furtheran approximation by the Ritz method: minimize Φ on Vk to get some uk ∈ Vk

satisfying the Galerkin identity (2.8) with A = Φ′. Show that uk ⇀ u where uminimizes Φ over V = W 1,p(Ω).25 Assume a subcritical growth of b(x, ·) and c(x, ·)and deduce the strong convergence (in terms of subsequences)26

uk → u in W 1,p(Ω). (4.40)

24Hint: Realize that, for p ≥ 2, the coefficient σ is nondecreasing and positive (hence (4.37)holds trivially) while, for p ≤ 2, (4.37) can be verified by calculations.

25Hint: Combine Exercise 4.21 with density of⋃

k∈NVk in W 1,p(Ω) as in Example 2.67.

26Hint: Use Exercise 4.21 with Φ1(u) = 1p

∫Ω |∇u(x)|pdx and then just use Theorem 1.2 and

uniform convexity of Lp(Ω;Rn).


Exercise 4.28 (Nonmonotone terms with critical growth). Consider the equation−Δu+c(u) = g with c(r) = r5−r2 in Ω ⊂ R3 with Dirichlet boundary conditions,n = 3, and show existence of a weak solution in W 1,2(Ω).27

Remark 4.29 (Strong convergence of Ritz’ method). In fact, only a strict convexityof the nonlinearity s �→ a(x, s) is sufficient for (4.40).28 This is a nontrivial effectthat, in this concrete potential case, the d-monotonicity needed in the abstractnon-potential case, cf. Remark 2.21, can be considerably weakened.

Example 4.30 (Advection �v · ∇u does not have any potential). Following Ex-ercise 2.91, we consider A : W 1,2(Ω) → W 1,2(Ω)∗ defined by 〈A(u), v〉 =∫Ω(�v ·∇u) vdx with a given vector field �v with, say, div�v = 0 and �v|Γ = 0. UsingGreen’s formula, we can evaluate

∫ 1

0

⟨A(tu), u

⟩dt =

∫ 1

0

∫Ω

tu�v·∇u dxdt =

∫ 1

0

t

∫Ω

�v·∇u2

2dxdt = −

∫Ω

(div �v)u2

4dx = 0.

By (4.6), a potential Φ of A would have to be constant so that Φ′ = 0, butobviously A �= 0. This shows that A cannot have any potential. Realize that, ofcourse, the condition (4.21) indeed fails.

Exercise 4.31 (Anisotropic p-Laplacean). Consider Φ(u) := 1p

∫Ω

∑ni=1 | ∂

∂xiu|pdx

on V := W 1,p0 (Ω). Show that

Φ′(u) = −n∑

i=1

∂

∂xi

(∣∣∣ ∂u∂xi

∣∣∣p−2 ∂u

∂xi

)= (p−1)

n∑i=1

∣∣∣ ∂u∂xi

∣∣∣p−2 ∂2u

∂x2i

and that Φ′ is monotone and, if p ≥ 2, uniformly monotone.29

Exercise 4.32 (Higher-order Euler-Lagrange equation). Consider the 4th-orderequation as in Exercise 2.98, i.e.

div div(a(x, u,∇u,∇2u)

)− div(b(x, u,∇u,∇2u)

)+ c(x, u,∇u,∇2u) = g (4.41)

here naturally with (a, b, c) : Ω×R×Rn×Rn×n → Rn×n×Rn×R, and show that

27Hint: Realize that 2∗ = 6 for n = 3 and combine convex continuous functional Φ1(u) :=16

∫Ωu6dx with nonconvex but weakly continuous functional Φ2(u) := − 1

3

∫Ωu3dx on W 1,2(Ω),

and realize coercivity of Φ(u) := 12

∫Ω |∇u|2dx+ Φ1(u) + Φ2(u).

28The proof, however, is rather nontrivial and uses so-called Young measures generated byminimizing sequences (here {uk}k∈N) which must be composed from Dirac measures if a(x, ·) isstrictly convex; cf. Pedregal [331, Theorem 3.16]. See also Visintin [417].

29Hint: Modify (2.139). For (uniform) monotonicity, modify (2.141) or (2.142).


the symmetry condition like (4.21) now looks as

∂aij(x, r, s, S)

∂Skl=

∂akl(x, r, s, S)

∂Sij,

∂aij(x, r, s, S)

∂Rk=

∂bk(x, r, s, S)

∂Sij, (4.42a)

∂aij(x, r, s, S)

∂r=

∂c(x, r, s, S)

∂Sij,

∂bi(x, r, s, S)

∂Rj=

∂bj(x, r, s, S)

∂Ri, (4.42b)

∂bi(x, r, s, S)

∂r=

∂c(x, r, s, S)

∂Ri, (4.42c)

for i, j, k, l=1, . . . , n, i.e. symmetry of the Jacobian of the mapping

(c(x, ·, ·, ·), b(x, ·, ·, ·), a(x, ·, ·, ·)) : R×Rn×Rn×n → R×Rn×Rn×n,

and then the potential is∫Ω ϕ(x, u,∇u,∇2u) dx with ϕ given as in (4.23b) now by

ϕ(x, r, s) =

∫ 1

0

S :a(x, tr, ts, tS) + s·b(x, tr, ts, tS) + r c(x, tr, ts, tS) dt . (4.43)

Exercise 4.33 (p-biharmonic operator). Consider aij in the previous Exercise 4.32given by (2.113) and bi = c = 0, verify (4.42), and evaluate (4.43) to show that thep-biharmonic operator Δ(|Δ|p−2Δ) on V := W 2,p

0 (Ω) has the potential Φ(u) :=1p

∫Ω|Δu|pdx. For p = 2, consider also Φ(u) = 1

2

∫Ω|∇2u|2dx.30

Exercise 4.34 (Singular perturbations). Consider the problem (2.167), use the es-timates (2.168) and the potentiality of the operator ε div2∇2 −Δp, and show theweak convergence by passing to the limit in the underlying minimization prob-lem.31 Modify it by considering a quasilinear regularizing term as in Example 2.46.


Calculus of variations and related variational problems have been cultivated in-tensively since the 17th century by Fermat, Newton, Leibniz, Bernoulli, Euler,

30Hint: Realize that ∂aij/∂Skl = 0 for i �= j or k �= l, and that (4.43) gives ϕ(S) =|∑n

k=1 Skk|p/p. Further realize, for p = 2, that∫Ω|∇2u|2dx =

∫Ω|Δ|2dx under the Dirich-

let boundary conditions, cf. Example 2.46.31Hint: Using that uε minimizes the functional u �→ ∫

Ω1p|∇u|p + ε

2|∇2u|2 − gu dx and that it

converges weakly to some u in W 1,p(Ω), show∫Ω

1

p|∇u|pdx ≤ lim inf

ε→0

∫Ω

1

p|∇uε|pdx ≤ lim inf

ε→0

∫Ω

1

p|∇uε|p +

ε

2|∇2uε|2dx

≤ limε→0

∫Ω

1

p|∇v|p +

ε

2|∇2v|2 − g(v−uε) dx =

∫Ω

1

p|∇v|p − g(v−u) dx

for any v ∈ W 2,20 (Ω) ∩W 1,p(Ω). By continuity,

∫Ω

1p|∇u|p − gu dx ≤ ∫

Ω1p|∇v|p − gv dx holds

for any v ∈ W 1,p0 (Ω).


Lagrange, Legendre, or Jacobi, often related to direct applications in physics andalways bringing inspiration to development of mathematics.

The exposition here is narrowly focused on coercive problems leading toelliptic boundary-value problems. As to the abstract theory presented in Sect. 4.1,for further reading we refer to Blanchard, Bruning [53, Chap.2,3], Dacorogna [112,Chap.3], Gajewski, Groger, Zacharias [168, Sect.III.4], Vainberg [414, Chap.II-IV],Zeidler [427, Parts II/B & III]. Reading about the problems having the potentialof the type

∫Ωϕ(u,∇u) dx may include in particular Dacorogna [112, Chap.3],

Evans [138, Chap.8], Gilbarg and Trudinger [Sect.11.5][178], Jost and Li-Jost [218],Ladyzhenskaya and Uraltseva [250, Chap.5].

Vectorial problems leading to systems of equations requiring special tech-niques, cf. also Sect. 6.1, are addressed e.g. by Dacorogna [112, Chap.IV], Evans[138, Chap.8], Giaquinta, Modica and J. Soucek [177, Part II, Sect.1.4], Giusti[180, Chap.5], Morrey [291], Muller [297], and Pedregal [331, Chap.3].

There are many other variational techniques relying on critical points dif-ferent from the global minimizers used here and more sophisticated principles,sometimes able to cope also with side conditions. Let us mention the celebratedmountain-pass technique by Ambrosetti and Rabinowitz [17] or Lyusternik andSchnirelman theory [263]. The monographs devoted to such advanced techniquesare, e.g., Blanchard and Bruning [53], Chabrowski [91], Fucık, Necas, Soucek [158],Giaquinta and Hildebrandt [176], Giaquinta, Modica and J. Soucek [177], Giusti[180], Kuzin and Pohozaev [247], Struwe [400], and Zeidler [427, Part III].

Chapter 5

Nonsmooth problems;variational inequalities

Many problems in physics and in other applications cannot be formulated as equa-tions but have some more complicated structure, usually of a so-called comple-mentarity problem. From the abstract viewpoint, the equations are replaced byinclusions involving set-valued mappings. We confine ourselves to a rather simplecase (but still having wide applications) which involves set-valued mappings whose“set-valued part” can be described as a subdifferential of a convex but nonsmoothpotential. Recall that we consider, if not said otherwise, V reflexive.

5.1 Abstract inclusions with a potential

A set-valued mapping A : V ⇒ V ∗ is called monotone if, for all f1 ∈ A(u1) andf2 ∈ A(u2), it holds that 〈f1−f2, u1−u2〉 ≥ 0. We admit A(u) = ∅ and denote thedefinition domain of A by dom(A) := {u ∈ V ; A(u) �= ∅}. Naturally, A : V ⇒ V ∗ iscalledmaximal monotone if the graph ofA is maximal (with respect to the orderingby inclusion) in the class of monotone graphs (i.e. graphs of monotone set-valuedmappings) in V × V ∗. By the Kuratowski-Zorn lemma, any monotone set-valuedmapping admits a maximal monotone extension, cf. Figure 10a,b. Besides, we callA : V ⇒ V ∗ coercive if

lim‖u‖→∞

inff∈A(u)

〈f, u〉‖u‖ = +∞. (5.1)

Here we shall consider some functional (again called a potential) Φ : V →R := R ∪ {+∞,−∞} such that the (set-valued) mapping A represents a certain


134 Chapter 5. Nonsmooth problems; variational inequalities

a) b) c)

1

1 1 −1

−1 −1

A(u) A(u)A(u)

uuu

Figure 10. a) a monotone but not maximal monotone mapping A : R → R,b) a maximal monotone extension A : R ⇒ R of the mapping from a),c) another maximal monotone A : R ⇒ R, inverse to the mapping from b);

note that it is a normal-cone mapping to the interval [−1, 1], cf. (5.3).

generalization of the gradient of Φ. We call Φ : V → R proper if it is not iden-tically equal to +∞ and does not take the value −∞. Except Remark 5.8, weconfine ourselves to the case that Φ is convex, and then A : V ⇒ V ∗ will be thesubdifferential of Φ, i.e. A = ∂Φ, defined by

∂Φ(u) :={f ∈V ∗; ∀v∈V : Φ(v) + 〈f, u− v〉 ≥ Φ(u)

}, (5.2)

cf. Figure 11. It is indeed a generalization of the gradient because, if A is also theGateaux differential, then ∂Φ(u) = {Φ′(u)}.1 If Φ is finite and continuous at u,then ∂Φ(u) �= ∅,2 otherwise emptiness of ∂Φ(u) is possible not only on V \dom(Φ)but also on dom(Φ) as well as situations when dom(∂Φ) is not closed, cf. Figure 11.

anothersupporti

ng hyperplane

a supportinghyperplane

0

1

1

1

+∞

u

u1 u2

−f1

f2f3

Φ

epi(Φ)

Figure 11. Subdifferential of a convex lower semicontinuous function; an exam-ple for ∂Φ(u1) = ∅, ∂Φ(u2) = [f1, f2] f3, and dom(Φ)=[u1,+∞)= dom(∂Φ)=(u1,+∞) because limu↘u1 Φ

′(u) = −∞.

Example 5.1 (Normal-cone mapping). If K is a closed convex subset of V andΦ(u) = δK(u) is the so-called indicator function, i.e. δK(u) = 0 if u ∈ K andδK(u) = +∞ if u �∈ K, then, by the definition (5.2), we can easily see that

∂δK(u) = NK(u) :=

{ {f∈V ∗; ∀v∈K : 〈f, v − u〉 ≤ 0

}for u∈K,

∅ for u �∈K,(5.3)

where NK(u) is the normal cone to K at u; cf. Figure 1 on p. 6.

Example 5.2 (Potential of the duality mapping). For Φ(u) = 12‖u‖2, it holds

∂Φ(u) = J(u), the duality mapping.3 For V ∗ strictly convex, cf. Example 4.19.

1Cf. Exercise 5.34 below.2This can be proved by the Hahn-Banach Theorem 1.5.3The inclusion ∂Φ(u) ⊃ J(u) follows from 1

2‖v‖2 − 1

2‖u‖2 ≥ ‖v‖ ‖u‖ − ‖u‖2 ≥ 〈f, v − u〉

for f ∈ J(u), cf. also (4.32). Conversely, f ∈ J(u) implies 12‖u + th‖2 − 1

2‖u‖2 ≥ t〈f, h〉. Then,

likewise (4.33), DΦ(u, h) ≥ 〈f, h〉 for any h ∈ X, hence inevitably f ∈ ∂Φ(u).

5.1. Abstract inclusions with a potential 135

Theorem 5.3 (Convex case4). Let A : V ⇒ V ∗ have a proper convex potentialΦ : V → R, i.e. A = ∂Φ. Then:

(i) A is closed-valued, convex-valued, and monotone.

(ii) If Φ is lower semicontinuous, then A is maximal monotone.

(iii) If Φ is also coercive, then A is surjective in the sense that the inclusion

A(u) � f (5.4)

has a solution for any f ∈ V ∗.

Proof. Closedness and convexity of the set ∂Φ(u) is obvious. To show monotonicityof the mapping ∂Φ, we use the definition (5.2) so that, for any fi ∈ ∂Φ(ui) withi = 1, 2, one has

Φ(u2) ≥ Φ(u1) + 〈f1, u2−u1〉 and Φ(u1) ≥ Φ(u2) + 〈f2, u1−u2〉. (5.5)

By a summation, one gets 〈f1 − f2, u1 − u2〉 ≥ 0.As to (ii), take (u0, f0)∈V ×V ∗ and assume that 〈f0 − f, u0 − u〉 ≥ 0 for any

(f, u)∈Graph(A). As V is assumed reflexive, we can consider it, after a possiblerenorming due to Asplund’s theorem, as strictly convex together with its dual.Then, we consider (f, u) such that J(u) + f = J(u0) + f0, f ∈A(u), J : V → V ∗

the duality mapping, i.e. [J+A](u) � J(u0)+f0; such u does exist due to the point(iii) below applied to the convex coercive functional v �→ 1

2‖v‖2 + Φ(v), cf. alsoExample 4.19. Then 0 ≤ 〈f0−f, u0−u〉 = 〈J(u)−J(u0), u0−u〉 and, by the strictmonotonicity of J , cf. Lemma 3.2(iv), we get u0 = u so that (f0, u0) ∈ Graph(A).

The point (iii) can be proved by the direct method: Φ convex and lowersemicontinuous implies that Φ is weakly lower semicontinuous; cf. Exercise 5.30.Then, by coercivity of Φ and reflexivity of V , the functional Φ − f , being alsocoercive, possesses a minimizer u, see Theorem 4.2. Then ∂Φ(u) � f becauseotherwise ∂Φ(u) �� f would imply, by the definition (5.2), that

∃v∈V : Φ(v) + 〈f, u− v〉 < Φ(u) (5.6)

so that [Φ−f ](v) = Φ(v)−〈f, v〉 < Φ(u)−〈f, u〉 = [Φ−f ](u), a contradiction. �Theorem 5.4 (Special nonconvex case). Let Φ = Φ1+Φ2 : V → R be coercive,Φ1 be a proper convex lower semicontinuous functional and Φ2 be a weakly lowersemicontinuous and Gateaux differentiable functional, and let A1 = ∂Φ1 and A2 =Φ′2. Then, for any f ∈ V ∗, there is u ∈ V solving the inclusion

A1(u) +A2(u) � f . (5.7)

Remark 5.5 (Alternative formulations: inequalities). The inclusion (5.7), writtenas ∂Φ1(u) � f − A2(u), represents, in view of (5.2), a problem involving thevariational inequality

Find u∈V : ∀v∈V : Φ1(v) +⟨A2(u), v − u

⟩ ≥ Φ1(u) + 〈f, v − u〉. (5.8)

4In fact, (i)-(ii) holds even for non-reflexive spaces; see Rockafellar [352].


Proof of Theorem 5.4. Coercivity and weak lower semicontinuity of Φ with reflex-ivity implies the existence of a minimizer u of Φ − f . In particular, Φ(u) < +∞and hence also Φ1(u) < +∞.

Suppose that (5.7) does not hold, i.e. ∂Φ1(u) �� f − Φ′2(u). By negation of(5.8), this just means that

∃v ∈ V : Φ1(v) + 〈f − Φ′2(u), u− v〉 < Φ1(u). (5.9)

For 0 < ε ≤ 1, put vε = u + ε(v−u). As Φ1 is convex, it has the directionalderivative DΦ1(u, v − u) and

DΦ1(u, v − u) := limε↘0

Φ1(vε)− Φ1(u)

ε= inf

ε>0

Φ1(vε)− Φ1(u)

ε

≤ Φ1(v1)− Φ1(u) = Φ1(v) − Φ1(u) < +∞. (5.10)

Note that DΦ1(u, v−u) is finite because it is bounded from below by −DΦ1(u, u−v) > −∞ by similar argument as (5.10). In particular,

Φ1(vε) = Φ1(u) + εDΦ1(u, v − u) + o1(ε) (5.11)

with some o1 such that limε↘0 o1(ε)/ε is 0. Moreover, as Φ′2 is smooth, hence〈Φ′2(u), v − u〉 = DΦ2(u, v − u), by the definition of Gateaux differential, it holdsthat

Φ2(vε) = Φ2(u) + ε〈Φ′2(u), v − u〉+ o2(ε) (5.12)

with some o2 such that limε→0 o2(ε)/ε = 0. Thus, adding (5.11) and (5.12) andusing (5.10), we get

Φ1(vε) + Φ2(vε)− 〈f, vε〉 = Φ1(u) + Φ2(u)− 〈f, u〉+ ε(DΦ1(u, v−u) + DΦ2(u, v−u)− 〈f, v−u〉

)+ o1(ε) + o2(ε)

≤ Φ1(u) + Φ2(u)− 〈f, u〉+ ε(Φ1(v)− Φ1(u) + 〈Φ′2(u), v−u〉 − 〈f, v−u〉

)+ o1(ε) + o2(ε). (5.13)

By (5.9), the multiplier of ε is negative, and therefore this term dominates o1(ε)+o2(ε) if ε>0 is sufficiently small. Hence, for a small ε>0, the functional Φ1+Φ2−ftakes at vε a lower value than at u, a contradiction. �Remark 5.6 (Special cases). If Φ1 := Φ0 + δK with both Φ0 : V → R and K ⊂ Vconvex, then, for A = A2, (5.8) turns into the variational inequality:

Find u∈K : ∀v∈K :⟨A(u), v−u

⟩+Φ0(v) − Φ0(u) ≥ 〈f, v−u〉. (5.14)

Often, Φ0 = 0 and then (5.14) can equally be written in the frequently used form

f −A(u) ∈ NK(u), (5.15)

which is a special case of (5.7).

5.2. Application to elliptic variational inequalities 137

Corollary 5.7. Let A = A1 + A2 : V ⇒ V ∗ have the set-valued part A1 : V ⇒ V ∗

monotone, coercive, and possessing a proper weakly lower semicontinuous potentialΦ1, and the single-valued part A2 : V → V ∗ be pseudomonotone and possessing a(smooth) potential Φ2 with an affine minorant. Then, for any f ∈ V ∗, the inclusion(5.4) has a solution.

Proof. Denote Φ1 and Φ2 the respective potentials. Then A1 coercive and mono-tone implies Φ1 coercive; cf. Exercise 5.32. Moreover, as Φ2 has an affine minorant,Φ1+Φ2 is also coercive. Furthermore, Φ2 is weakly lower semicontinuous, see The-orem 4.4(ii). Then we can use Theorem 5.4. �

Remark 5.8 (Hemivariational inequalities). In case of a general nonconvex locallyLipschitz Φ, the so-called Clarke (generalized) gradient is defined by:

∂CΦ(u) :={f ∈V ∗; ∀v∈V : D◦Φ(u, v) ≥ 〈f, v〉

}(5.16)

where D◦Φ(u, v) denotes the generalized directional derivative defined by

D◦Φ(u, v) := lim supu→uε↘0

Φ(u+εv)− Φ(u)

ε; (5.17)

see Clarke [96] for more details. Inclusions involving Clarke’s gradients are calledhemivariational inequalities. In the special case of Theorem 5.4, we have ∂CΦ =∂Φ1 +Φ′2 provided Φ1 is locally Lipschitz continuous.

5.2 Application to elliptic variational inequalities

We will illustrate the previous theory on the 2nd-order elliptic variational in-equality forming a so-called unilateral problem with an obstacle (determined by afunction w) distributed over Ω and with Newton-type boundary conditions:

−div a(x, u,∇u) + c(x, u,∇u) ≥ g ,

u ≥ w,(div a(x, u,∇u)− c(x, u,∇u) + g

)(u − w) = 0

⎫⎪⎬⎪⎭ in Ω,

ν · a(u,∇u) + b(x, u) ≥ h,

u ≥ w,(ν · a(x, u,∇u) + b(x, u,∇u)− h

)(u− w) = 0

⎫⎪⎬⎪⎭ on Γ.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭(5.18)

The equalities in (5.18) express transversality of residua from the correspond-ing inequalities, while the triple composed from these two inequalities and onetransversality relation is called a complementarity problem.


An interpretation illustrated in Figure 12 in a two-dimensional case is thatu is a vertical deflection of an elastic membrane5 elastically fixed on the contourΓ and stretched above a nonpenetrable obstacle given by the graph of w.

a contact zone

a free boundary

a membrane

an obstacle

u,w

u = u(x1, x2)

w = w(x1, x2)

x1x2

Γ+

Ω+ Ω0

Figure 12. A schematic situation of unilateral problems on Ω ⊂ R2: a deflectedelastic membrane being in a partial contact with a rigid obstacle.

The abstract inequality (5.14) with Φ0 = 0, A given by (2.59) and f by (2.60)leads to the weak formulation, resulting in a variational inequality:

Find u∈K : ∀v∈K:

∫Ω

a(u,∇u) · ∇(v−u) + c(u,∇u)(v−u) dx

+

∫Γ

b(u)(v−u) dS ≥∫Ω

g(v−u) dx+

∫Γ

h(v−u) dS (5.19)

whereK :=

{v∈W 1,p(Ω); v ≥ w in Ω

}. (5.20)

Assuming the symmetry condition (4.21), in view of Lemma 4.13, we can con-sider the abstract inequality (5.14) alternatively with A = 0, f from (2.60), andΦ0 defined as in (4.23a) and ϕ and ψ are defined by (4.23b,c), which results inminimization over W 1,p(Ω) of the potential

Φ− f : u �→ Φ0(u)− 〈f, u〉+ δK(u)

:=

∫Ω

(ϕ(u,∇u)− gu

)dx+

∫Γ

(ψ(u)− hu

)dS + δK(u). (5.21)

An important question is whether the weak formulation (5.19) is consistentand selective; in other words, whether (5.18) contains enough information and isnot overdetermined. The positive answer is:

Proposition 5.9 (Weak vs. classical formulations). Let w ∈ C(Ω) and u ∈C2(Ω). Then the inequality (5.19) is satisfied if and only if (5.18) holds.

Proof. Let us denote

Ω+ :={x∈Ω; u(x) > w(x)

}, Ω0 :=

{x∈Ω; u(x) = w(x)

},

Γ+ :={x∈Γ; u(x) > w(x)

}, Γ0 :=

{x∈Γ; u(x) = w(x)

},

5To be more precise, this interpretation refers to a(x, r, s) = αs with α > 0 the elasticitycoefficient, c = 0, and g a tangential outer force per unit area.


cf. Figure 12. For u solving (5.18) and for any v ∈ W 1,p(Ω) such that v ≥ w, byGreen’s formula, we can write∫

Ω

a(u,∇u) · ∇(v − u) + c(u,∇u)(v − u) dx+

∫Γ

b(u)(v − u) dS

=

∫Ω

(div a(u,∇u)− c(u,∇u)

)(u−v) dx +

∫Γ

(ν · a(u,∇u) + b(u)

)(v−u) dS

=

∫Ω+


)(u−v) dx

+

∫Ω0


)(u−v) dx

+

∫Γ+

(ν · a(u,∇u) + b(u)

)(v − u) dS +

∫Γ0

(ν · a(u,∇u) + b(u)

)(v − u) dS

=:

∫Ω+

I1(x) dx +

∫Ω0

I2(x) dx +

∫Γ+

I3(x) dS +

∫Γ0

I4(x) dS.

Now, by (5.18), we have I1 = g(v−u) in Ω+, I2 ≥ g(v−u) because div a(u,∇u)−c(u,∇u) ≤ −g and u − v = w − v ≤ 0 in Ω0, I3 = h(v − u) on Γ+, and finallyI4 ≥ h(v−u) because ν ·a(u,∇u)+ b(u) ≥ h and v−u = v−w ≥ 0 on Γ0. Hence,altogether∫

Ω+

I1 dx+

∫Ω0

I2 dx+

∫Γ+

I3 dS +

∫Γ0

I4 dS ≥∫Ω

g(v−u) dx+

∫Γ

h(v−u) dS

so that (5.19) has been obtained.

Conversely, if the solution u to (5.19) is regular enough, we can take z smoothsuch that supp(z) ⊂ Ω+. Then, for a sufficiently small |ε|, v := u+εz ∈ K so that,by putting it into (5.19), one gets∫

Ω

a(u,∇u) · ∇z + c(u,∇u)z dx ≥∫Ω

gz dx. (5.22)

Considering also −z instead of z, we get an equality in (5.22). Then, by using theGreen formula, we get div a(u,∇u)− c(u,∇u) + g = 0 a.e. in Ω+. The inequalityu ≥ w is directly involved in (5.19). Since, for z ≥ 0 with supp(z) ⊂ Ω0, alwaysv = u + z ∈ K, we get by putting such v into (5.19) the inequality

∫Ωgz dx ≤∫

Ω a(u,∇u) · ∇z + c(u,∇u)z dx =∫Ω(−div a(u,∇u) + c(u,∇u))z dx, which gives

div a(u,∇u)−c(u,∇u)+g ≤ 0 a.e. in Ω0. Altogether, the complementarity relationsin Ω constituting (5.18) have been verified.

The complementarity relations on Γ can be verified analogously by takingtest functions having nonvanishing traces on Γ. If u(x) > w(x) for some x ∈ Γ,then, taking a sufficiently small neighbourhood N of x, we have u > w on Ω ∩N(i.e. N ∩ Ω0 = ∅), and then v := u + εz ∈ K for a sufficiently small |ε| provided


supp(z) ⊂ Ω ∩N . Putting it into (5.19), one gets∫Ω∩N

a(u,∇u)·∇z+c(u,∇u)z dx+

∫Γ∩N

b(u)z dS ≥∫Ω∩N

gz dx+

∫Γ∩N

hz dS.

By using the Green formula, we get∫Ω∩N

(− div a(u,∇u) + c(u,∇u)− g)z dx

+

∫Γ∩N

(ν · a(u,∇u) + b(u)− h

)z dS ≥ 0. (5.23)

Considering also −z instead of z, we get equality in (5.23). As we already knowthat div a(u,∇u)−c(u,∇u)+g = 0 in Ω∩N ⊂ Ω+, we get ν ·a(u,∇u)+b(u)−h = 0on Γ∩N . In general, we can take z ≥ 0 arbitrary but such that it will be small inΩ yet still with prescribed values on Γ. This will push the first integral in (5.23)to zero while the second one then yields ν · a(u,∇u) + b(u)− h ≥ 0 on Γ. �

A theoretically and to some extent also numerically6 efficient method of reg-ularization (or approximation) for problems like (5.18) is the so-called penaltymethod. In the potential case, its Lα-variant leads to approximation of the func-tional from (5.21) by the functional

Φε(u) =

∫Ω

(ϕ(u,∇u) +

((w − u)+

)ααε

)dx+

∫Γ

ψ(u) dS (5.24)

where v+ := max(0, v). The idea is then to minimize Φε − f over the wholeW 1,p(Ω), which corresponds to the boundary-value problem

−div a(u,∇u) + c(u,∇u)− 1

ε

((w − u)+

)α−1= g in Ω,

ν · a(u,∇u) + b(x, u) = h on Γ.

⎫⎬⎭ (5.25)

Now, we need to modify the coercivity 〈A(v), v−w〉 ≥ δ‖v‖pW 1,p(Ω)+C with δ > 0

and some C (depending possibly on w). E.g., we can modify (4.31) to

a(x, r, s)·(s−∇w(x)) + c(x, r, s)(r−w(x))

≥ ε1|s|p + ε2|r|q − k0(x)− k1(x)|s| − k2(x)|r|, (5.26a)

b(x, r)(r−w(x)) ≥ −k3(x)− k4(x)|r| (5.26b)

with some ε0, ε1 > 0, p ≥ q > 1, and k0 ∈ L1(Ω), k1 ∈ Lp′(Ω), k2 ∈ Lp∗′

(Ω),

k3∈L1(Γ), and k4∈Lp#′(Γ) depending on a fixed w.

6If also a discretization as in Exercise 4.27 is applied, one can implement the resulting min-imization problem on computers although its numerical solution is not always easy if ε > 0 in(5.24) has to be chosen small.


Proposition 5.10 (Convergence of the penalty method). Assume 1 < α ≤p∗, and a, b and c satisfy the qualifications in Lemmas 4.13–4.14, in particular thesymmetry (4.21) and coercivity in the sense (5.26) hold. Let also w ∈ W 1,p(Ω).Then:

(i) The boundary-value problem (5.25) has always a weak solution uε ∈ W 1,p(Ω).

(ii) The sequence {uε}ε>0 is bounded in W 1,p(Ω). If Φ0 is convex, then thevalues of Φ0 converge, i.e. limε→0 Φ0(uε) := limε→0

∫Ωϕ(uε,∇uε) dx +∫

Γ ψ(uε) dS → ∫Ω ϕ(u,∇u) dx+

∫Γ ψ(u) dS =: Φ0(u), and {uε}ε>0 converges

for ε → 0 (in terms of subsequences) weakly to a solution u of (5.19).

(iii) If, in addition, the mapping A induced by (a, c) is d-monotone with respect tothe seminorm v �→ ‖∇v‖Lp(Ω;Rn), then uε → u (a subsequence) in W 1,p(Ω)strongly.

Proof. For ε > 0 fixed, existence of a weak solution uε ∈ W 1,p(Ω) to (5.25) followsby the direct method by using Proposition 4.16.

To prove a-priori estimates, put v := uε − w into the weak formulation of(5.25). Using also (5.26) and the estimates as in the proof of Lemma 4.15, for asuitable δ and C, one gets

δ∥∥uε

∥∥qW 1,p(Ω)

+1

ε

∥∥(w−uε)+∥∥αLα(Ω)

− C

≤∫Ω

a(uε,∇uε) · ∇(uε−w) + c(uε,∇uε)(uε−w) +|(w−uε)

+|αε

dx

+

∫Γ

b(uε)(uε − w) dS =

∫Ω

g(uε − w) dx +

∫Γ

h(uε − w) dS. (5.27a)

The integrals on the right-hand side can be estimated as∫Ω

g(uε − w) dx +

∫Γ

h(uε − w) dS =⟨f, uε − w

⟩≤ ∥∥f∥∥

W 1,p(Ω)∗(∥∥uε

∥∥W 1,p(Ω)

+∥∥w∥∥

W 1,p(Ω)

)≤ δ

2

∥∥uε

∥∥qW 1,p(Ω)

+ Cδ

∥∥f∥∥q′W 1,p(Ω)∗ +

∥∥f∥∥W 1,p(Ω)∗

∥∥w∥∥W 1,p(Ω)

(5.27b)

with f determined by (and estimated through) the data (g, h), cf. (2.60) and (2.62),and with Cδ = q′ q−1

√qδ/2, cf. (1.22). In this way, we show {uε}ε>0 bounded in

W 1,p(Ω) and, up to a subsequence, uε ⇀ u in W 1,p(Ω).We have also 1

ε‖(w−uε)+‖αLα(Ω) ≤ C so that ‖(w−uε)

+‖Lα(Ω) = O( α√ε).

Using the weak continuity of v �→ ‖v+‖Lα(Ω) if α < p∗ (or the weak lower semi-continuity if α = p∗, cf. Exercise 5.30 below), one gets

‖(w−u)+‖Lα(Ω) ≤ lim infε→0

‖(w−uε)+‖Lα(Ω) ≤ lim sup

ε→0‖(w−uε)

+‖Lα(Ω) = 0.

Thus u ∈ K has been proved.


The convergence of Φ0(uε) to Φ0(u) can be seen from the estimate[Φ0−f

](u) ≤ lim inf

ε→0

[Φ0−f

](uε) ≤ lim sup

ε→0

[Φ0−f

](uε) ≤

[Φ0−f

](u) (5.28)

because Φ0 is weakly lower semicontinuous (see Lemma 4.14) and always[Φ0−f ](uε) ≤ [Φε−f ](uε) ≤ [Φε−f ](u) = [Φ0−f ](u) because uε minimizes Φε−fwith Φε := Φ0 + 1

αε‖(w − ·)+‖αLα(Ω) and because u ∈ K; here we used the as-sumption that Φ0 is convex so that each solution to the Euler-Lagrange equationminimizes its potential Φ0−f . Then (5.28) yields limε→0[Φ0−f ](uε) = [Φ0−f ](u),from which limε→0 Φ0(uε) = Φ0(u) follows.

The fact that u solves (5.19), i.e. minimizes (5.21), follows directly from theproved facts that u ∈ K and [Φ0−f ](u) ≤ min(Φ0 − f + δK), proved in (5.28) ifone realizes also

[Φ0−f ](uε) ≤ [Φε−f ](uε) ≤ min[Φε−f ] ≤ min(Φ0 − f + δK) (5.29)

because uε minimizes Φε−f and because always Φε ≤ Φ0 + δK .Now, we are going to prove the strong convergence. By multiplying the equa-

tion in (5.25) by (u − uε), applying Green’s formula and using the boundaryconditions in (5.25), one gets∫

Ω

a(uε,∇uε) · ∇(u−uε) + c(uε,∇uε)(u−uε)− 1

ε

∣∣(w−uε)+∣∣α−1

(u−uε) dx

+

∫Γ

b(uε)(u−uε) dS =

∫Ω

g(u−uε) dx +

∫Γ

h(u−uε) dS. (5.30)

Since u ≥ w, the term 1ε |(w−uε)

+|α−1(u−uε) ≥ 0 a.e. on Ω and, by omitting it,one gets∫

Ω

a(uε,∇uε) · ∇(u−uε) + c(uε,∇uε)(u−uε) dx

+

∫Γ

b(uε)(u−uε) dS ≥∫Ω

g(u−uε) dx+

∫Γ

h(u−uε) dS. (5.31)

Then, if (a, c) induces a d-monotone mapping as assumed, by (5.31) we get(d(‖∇uε‖Lp(Ω;Rn)

)− d(‖∇u‖Lp(Ω;Rn)

))(‖∇uε‖Lp(Ω;Rn) − ‖∇u‖Lp(Ω;Rn)

)≤∫Ω

(a(uε,∇uε)−a(u,∇u)) · ∇(uε−u) + (c(uε,∇uε)−c(u,∇u))(uε−u) dx

≤∫Ω

(g − c(u,∇u)

)(uε−u)− a(u,∇u) · ∇(uε−u) dx

+

∫Γ

(h− b(uε)

)(uε−u) dS → 0, (5.32)

because subsequently uε − u ⇀ 0 in Lp∗(Ω), h− b(uε) → h− b(u) in Lp#′

(Γ) and

uε−u ⇀ 0 in Lp#

(Γ), and ∇uε −∇u ⇀ 0 in Lp(Ω;Rn). From (5.32), one deduces


‖∇uε‖Lp(Ω;Rn) → ‖∇u‖Lp(Ω;Rn). From ∇uε ⇀ ∇u in Lp(Ω;Rn) proved above, andfrom the uniform convexity of Lp(Ω;Rn), by Theorem 1.2 we get ∇uε → ∇u. �

In case Φ0 is not convex, weak solutions uε to the boundary-value problem(5.25) do not necessarily minimize Φε−f . Anyhow, the proof of the strong conver-gence in Proposition 5.10(iii) holds, while the proof of mere weak convergence isto be made through the Minty trick. For example, if (c, a) is monotone but b not(hence Φ0 indeed need not be convex), one can prove:

Proposition 5.11 (Convergence of the penalty method II). Let 1<α≤p∗,and a, b and c satisfy the qualifications in Lemmas 4.13–4.14 and let (c, a) inducea monotone mapping (4.22) for a.a. x ∈ Ω. Then uε from Proposition 5.10(i)converges for ε → 0 (in terms of subsequences) weakly to a solution u of (5.19).

Proof. We now want to use the Minty trick. To this goal, we multiply (5.25) by (v−uε), apply Green’s formula, and use the boundary conditions in (5.25) to get (5.30)with v in place of u. Considering v ≥ w, one can see that 1

ε |(w−uε)+|α−1(v−uε) is

non-negative a.e. on Ω. Thus we arrive at (5.31) with v in place of u. Then, usingmonotonicity of [c, a](x, ·, ·), we obtain

0 ≤∫Ω

(a(uε,∇uε)−a(v,∇v))·∇(uε−v) + (c(uε,∇uε)−c(v,∇v))(uε−v) dx

≤∫Ω

(g − c(v,∇v)

)(uε−v)− a(v,∇v)·∇(uε−v) dx+

∫Γ

(h− b(uε)

)(uε−v) dS

→∫Ω

(g−c(v,∇v)

)(u−v)− a(v,∇v)·∇(u−v) dx+

∫Γ

(h−b(u)

)(u−v) dS;

(5.33)

here we used compactness of the trace operator W 1,p(Ω) → Lp#−ε(Γ) for b(uε) →b(u) in Lp#′

(Γ). Then we modify Minty’s trick: instead of v := u + ηz used forequations with η > 0, cf. the proof of Lemma 2.13, we now put v := ηz + (1−η)ufor η ∈ (0, 1] and z ≥ w; note that such v lives in K. As v−u = ηz+(1−η)u−u =η(z−u), this gives

0 ≤∫Ω

(g − c

(ηz+(1−η)u , η∇z+(1−η)∇u

))(ηu−ηv)

− a(ηz+(1−η)u , η∇z+(1−η)∇u

)·∇(ηz−ηu) dx+

∫Γ

(h−b(u)

)(ηz−ηu) dS.

Then we divide it by η > 0, and pass to the limit with η → 0. It gives just thedesired inequality (5.19). �

Remark 5.12. Strenghtening the growth condition (2.55) and the qualification of


w e.g. by considering w ∈ W 1,∞(Ω) and

|a(x, r, s)| ≤ γ(x) + C|r|q−ε + C|s|p−1 for some γ∈Lp′(Ω) , (5.34a)

| b(x, r) | ≤ γ(x) + C|r|q−ε for some γ∈Lp#′(Γ), (5.34b)

|c(x, r, s)| ≤ γ(x) + C|r|q−ε + C|s|p−ε for some γ∈Lp∗′(Ω) (5.34c)

and for some ε > 0, one can replace the optimal but rather cumbersome coercivitycondition (5.26) depending on w by the former condition (4.31). Then, instead of(5.27), we can estimate for some δ > 0:

δ∥∥uε

∥∥qW 1,p(Ω)

+1

ε

∥∥(w−uε)+∥∥αLα(Ω)

− C

≤∫Ω

ε1|∇uε|p + ε2|uε|q − k0|∇uε| − k1|uε|+ |(w−uε)+|α

εdx−

∫Γ

k2|uε| dS

≤∫Ω

a(uε,∇uε) · ∇uε + c(uε,∇uε)uε +|(w−uε)

+|αε

dx+

∫Γ

b(uε)uε dS

=

∫Ω

a(uε,∇uε)·∇w + c(uε,∇uε)w + g(uε−w) dx +

∫Γ

b(uε)w + h(uε−w) dS

≤ Cδ1 + δ1

∫Ω

|∇uε|p + |uε|q dx+∥∥f∥∥

W 1,p(Ω)∗(∥∥uε

∥∥W 1,p(Ω)

+∥∥w∥∥

W 1,p(Ω)

)for any small δ1 and Cδ1 depending on δ1 and γ’s and C from (5.34) with ε1,ε2, k0, k1, and k2 from (4.31). Then the a-priori estimates ‖uε‖W 1,p(Ω) ≤ C and‖(w−uε)

+‖Lα(Ω) ≤ C/ α√ε easily follow.

Remark 5.13 (Free boundary problems). The boundary Ω+ ∩ Ω0, which is notknown a-priori and is thus a part of the solution to (5.18), is called a free boundary,cf. Figure 12. We thus speak about free-boundary problems, abbreviated often asFBPs.

Remark 5.14 (Dual approaches). Having in mind the constraint u ≥ w as in (5.18),one can write δK used in (5.21) as

δK(u) = sup0≤λ∈Lp∗′(Ω)

∫Ω

(w − u)λdx. (5.35)

Then, defining the so-called Lagrangean L(u, λ) := Φ(u)− 〈f, u〉+ ∫Ω(w − u)λdx

and realizing that L(·, λ) is convex while L(u, ·) is concave, we have

minu∈K

[Φ− f ] = minu∈W 1,p(Ω)

(Φ(u)− 〈f, u〉+ sup

0≤λ∈Lp∗′(Ω)

∫Ω

(w − u)λdx)

= minu∈W 1,p(Ω)

sup0≤λ∈Lp∗′(Ω)

(Φ(u)− 〈f, u〉+

∫Ω

(w−u)λdx)

≥ sup0≤λ∈Lp∗′(Ω)

minu∈W 1,p(Ω)

L(u, λ);

5.3. Some abstract non-potential inclusions 145

the last inequality holds because always minv∈W 1,p(Ω) L(v, λ) ≤ L(u, λ) ≤sup0≤ξ L(u, ξ) = Φ(u) for any u and λ.7 Thus the problem now consists in seekinga saddle point of the Lagrangean L. The problem of finding a supremum over{λ ≥ 0} of the concave function

Ψ(λ) := minu∈W 1,p(Ω)

L(u, λ) (5.36)

is referred to as the dual problem and can sometimes be easier to solve or/and givesuseful additional information; e.g. the constraint in the dual problems are simplerand, having an approximate maximizer λ∗ ≥ 0 of Ψ and an approximate minimizeru∗ ≥ w of Φ, we have a two-sided estimate Ψ(λ∗) ≤ minu∈K [Φ − f ] ≤ Φ(u∗).Cf. Exercise 5.51 for a concrete case of Ψ.

5.3 Some abstract non-potential inclusions

In this section we will again come back to the abstract level and deal with theinclusion of the type

∂Φ(u) +A(u) � f (5.37)

with Φ convex and with A pseudomonotone but not necessarily having a poten-tial. We will thus generalize Corollary 5.7 for the case that the smooth part hasno potential. Simultaneously, we will illustrate a general-purpose regularizationtechnique for the nonsmooth part of (5.37).8

Theorem 5.15. Let Φ : V → R ∪ {+∞} be convex, lower semicontinuous, proper,and possess, for any ε > 0, a convex, Gateaux differentiable regularization Φε :V → R such that Φ′ε : V → V ∗ is bounded and radially continuous and Φε → Φin the sense

∀v ∈ V : lim supε→0

Φε(v) ≤ Φ(v), (5.38a)

vε ⇀ v =⇒ lim infε→0

Φε(vε) ≥ Φ(v), (5.38b)

and A : V → V ∗ be pseudomonotone (non-potential, in general) and let, for someζ : R+ → R+ such that lims→+∞ ζ(s) = +∞, the following uniform coercivityhold:

∃v∈domΦ ∀ε > 0 ∀u∈V :Φε(u) + 〈A(u), u− v〉

‖u‖ ≥ ζ(‖u‖). (5.39)

Then, for any f ∈ V ∗, there is at least one u ∈ V solving the inclusion (5.37).

7Let us remark that the opposite inequality would require a constraint qualification, herep > n so that K from (5.20) would have a nonempty interior.

8For a usage of the regularization to potential problems see also Proposition 5.10. In (5.25),one uses ‖·‖αLα -penalty term instead of ‖·‖2

W1,p implied by usage of the formula (5.49), however.


Proof. By the coercivity (5.39) and the previous results, see Theorem 2.6 withLemmas 2.9 and 2.11(i)9, the regularized problem possesses a solution uε ∈ V , i.e.

Φ′ε(uε) +A(uε) = f. (5.40)

Moreover, we show that the coercivity (5.39) is uniform with respect to ε, andhence uε will be a-priori bounded. Indeed, as Φε is convex, in view of (5.8), theequation (5.40) means equivalently

Φε(v) + 〈A(uε)− f, v − uε〉 ≥ Φε(uε). (5.41)

Moreover, for v ∈ dom(Φ) and ε > 0 small enough, Φε(v) ≤ Φ(v) + 1 by (5.38a).Using subsequently (5.39), (5.41), and Φε(v) ≤ Φ(v) + 1, we get the estimate

ζ(‖uε‖)‖uε‖ ≤ Φε(uε) +⟨A(uε), uε − v

⟩≤ Φε(v) + 〈f, uε − v〉 ≤ Φ(v) + 1 + ‖f‖∗

(‖uε‖+ ‖v‖). (5.42)

Hence, the sequence {uε}ε>0 is bounded and, after taking possibly a subsequence,we can assume uε ⇀ u.

Now, for v ∈ V arbitrary, we will pass to the limit in (5.41). The right-hand side of (5.41) can be estimated by (5.38b) while (5.38a) can be used for theleft-hand side to get:

Φ(v)− 〈f, v−u〉+ lim supε→0

〈A(uε), v−uε〉

≥ lim supε→0

(Φε(v) + 〈A(uε)−f, v−uε〉

)≥ lim inf

ε→0Φε(uε) ≥ Φ(u). (5.43)

Passing to the limit with ε → 0, we get Φ(v) + 〈A(u), v − u〉 ≥ 〈f, v − u〉+ Φ(u),which is just (5.37), provided we still prove

lim supε→0

〈A(uε), v − uε〉 ≤ 〈A(u), v − u〉. (5.44)

To do this, we use the pseudomonotonicity of A: we are then to verify

lim infε→0

〈A(uε), u− uε〉 ≥ 0. (5.45)

Using (5.41) for v := u, we have

〈A(uε), u− uε〉 ≥ 〈f, u− uε〉+Φε(uε)− Φε(u) (5.46)

so that, by using again (5.38),

lim infε→0

〈A(uε), u− uε〉 ≥ lim infε→0

(〈f, u− uε〉+Φε(uε)− Φε(u)

)≥ lim

ε→0〈f, u− uε〉+ lim inf

ε→0Φε(uε)

− lim supε→0

Φε(u) ≥ 0 + Φ(u)− Φ(u) = 0, (5.47)

which proves (5.45). �9The coercivity of Φ′

ε + A (even uniform in ε > 0) follows via the test of (5.40) by uε − vfrom (5.42) below.


Remark 5.16 (Mosco’s convergence). One can weaken (5.38a) to10

∀v∈V ∃vε → v =⇒ lim supε→0

Φε(vε) ≤ Φ(v) (5.48)

and then (5.38b) with (5.48) is called Mosco’s convergence [293] of Φε to Φ; cf. Ex-ercise 5.35 below. This is advantageous in particular if the regularization is com-bined with the Galerkin method.

A concrete regularization Φε of Φ can be obtained by the formula

Φε(u) := infv∈V

‖u− v‖22ε

+Φ(v); (5.49)

here Φε is called the Yosida approximation of the functional Φ. Note that, forΦ = δK , we have obviously Φε(u) = 1

2εdist(u,K)2. Realize the coincidence withthe penalty method (5.24) for q = 2 and for ‖ · ‖ being the L2-norm.

Lemma 5.17 (Yosida approximation). Let Φ : V → R be convex, proper, lowersemicontinuous. Then:

(i) Each Φε is convex and lower semicontinuous and the family {Φε}ε>0 approx-imates Φ in the sense (5.38).

(ii) If V and V ∗ are strictly convex and reflexive, then each Φε is Gateaux differ-entiable and the differential Φ′ε : V → V ∗ is demicontinuous and bounded.

Proof. Denote Ψε(u, v) =12ε‖u− v‖2 +Φ(v).

(i) The lower semicontinuity: take uk → u and consider a minimizer vk forΨε(uk, ·), i.e.11 ∥∥uk − vk

∥∥2 = 2ε(Φε(uk)− Φ(vk)

). (5.50)

As {Φε(uk)}k∈N is bounded from above12 and Φ, being proper, has an affine mino-rant, (5.50) implies that {vk}k∈N is bounded. Considering vk ⇀ v (a subsequence),by estimating the limit inferior in (5.50) we obtain

‖u− v‖22ε

+Φ(v) = limk→∞

‖uk − v‖22ε

+Φ(v) ≥ lim infk→∞

Φε(uk)

= lim infk→∞

‖uk − vk‖22ε

+Φ(vk) ≥ ‖u− v‖22ε

+Φ(v) (5.51)

10By weakening (5.38a) to ∃vε ⇀ v : lim supε→0 Φε(vε) ≤ Φ(v), we would get the so-called Γ-convergence. This would, however, not be sufficient for passing to the limit in (5.47).Instead of the recovery sequence for {Φε}ε>0 itself, we should rather use recovery sequencesfor {Φε}ε>0 together with {Aε}ε>0 in the spirit of [281], namely here we might impose thecondition: for all uε ⇀ u and v ∈ V there is a “mutual recovery sequence” {vε}ε>0 such thatlimsupε→0 Φε(vε) + 〈A(uε)−f, vε−uε〉 − Φε(uε) ≤ Φ(v) + 〈A(u)−f, v−u〉 − Φ(u). The limitpassage in (5.41) is then obvious.

11Existence of vk follows by coercivity of Ψε(uk, ·) (because Φ, being proper, has an affineminorant) and by its weak lower semicontinuity, cf. Exercise 5.30 and the proof of Theorem 4.2.

12Note that Φε(uk) ≤ ‖uk−w‖2/ε+Φ(w)→ ‖u−w‖2/ε+Φ(w) < +∞ for w ∈ dom(Φ) fixed.


for any v ∈ V . Hence v minimizes Ψε(u, ·) so that ‖u− v‖2/(2ε) + Φ(v) = Φε(u),which showes the lower semicontinuity of Φε.

We now prove that Φε is convex: taking u1, u2 ∈ V and v1 a minimizer forΨε(u1, ·) and v2 a minimizer for Ψε(u2, ·), we have

Φε

(u1 + u2

2

)= inf

v∈VΨε

(u1 + u2

2, v)≤ Ψε

(u1 + u2

2,v1 + v2

2

)≤ 1

2Ψε(u1, v1) +

1

2Ψε(u2, v2) =

1

2Φε(u1) +

1

2Φε(u2). (5.52)

By the obvious fact Φε ≤ Φ, (5.38a) immediately follows. To prove (5.38b),let us realize that Φε ≥ Φδ provided 0 < ε ≤ δ. Then, for vε ⇀ v and for anyδ > 0, the convexity and lower semicontinuity of Φδ implies

lim infε→0

Φε(vε) ≥ lim infε→0

Φδ(vε) ≥ Φδ(v). (5.53)

Now, (5.38b) follows if one shows limδ→0 Φδ(v) = Φ(v). First, let v ∈ dom(Φ). Letvδ be a minimizer for Ψδ(v, ·), i.e.

‖v − vδ‖2 = 2δ(Φδ(v)− Φ(vδ)

). (5.54)

As {Φδ(v)}δ>0 is bounded from above by Φ(v) < +∞ and Φ has an affine mi-norant, (5.54) implies {vδ}δ>0 bounded. Then one can claim that {Φ(vδ)}δ>0 isbounded from below, and then (5.54) gives vδ → v. By (5.54), always

Φ(v) ≥ Φδ(v) ≥ Φ(vδ). (5.55)

By the lower semicontinuity of Φ,

Φ(v) ≥ lim supδ→0

Φδ(v) ≥ lim infδ→0

Φδ(v) ≥ lim infδ→0

Φ(vδ) ≥ Φ(v), (5.56)

showing that limδ→0 Φδ(v) = Φ(v). Second, consider v �∈ dom(Φ). Assumelimδ→0 Φ(vδ) �= Φ(v), i.e. Φδ(v) ≤ C for some C < +∞. Yet, then (5.54) againgives vδ → v and (5.56) implies Φ(v) ≤ C, a contradiction.

(ii) Let uε be a solution to the minimization problem in (5.49). From theoptimality conditions, we get

1

εJ(uε − u) + ∂Φ(uε) � 0, (5.57)

cf. Examples 4.19 and 5.2. This gives uε = (I+εJ−1∂Φ)−1(u) and also Φ(uε) −Φ(w) ≤ 1

ε 〈J(u−uε), uε−w〉 for any w. In particular, considering some v, we willuse it for w := vε := (I+εJ−1∂Φ)−1(v) to estimate (while abbreviating uε = u−uε


and vε = v−vε):

Φε(u)− Φε(v) = Φ(uε)− Φ(vε) +‖uε‖22ε

− ‖vε‖22ε

≤⟨J(uε)

ε, uε − vε

⟩+

‖uε‖22ε

− ‖vε‖22ε

=⟨J(uε)

ε, u−v

⟩−⟨J(uε)

ε, uε − vε

⟩+

‖uε‖22ε

− ‖vε‖22ε

≤⟨J(uε)

ε, u−v

⟩− ‖uε‖2

2ε− ‖vε‖2

2ε+

‖uε‖ ‖vε‖ε

≤⟨J(uε)

ε, u−v

⟩. (5.58)

In particular, 1εJ(u−uε) ∈ ∂Φε(u). By the same arguments, Φε(v) − Φε(u) ≤

〈1εJ(u−uε), v−u〉. Denoting

Aε(u) :=1

εJ(u−uε), (5.59)

we obtain⟨Aε(v)− Aε(u), v − u

⟩ ≥ Φε(v)− Φε(u)−⟨Aε(u), v − u

⟩ ≥ 0, (5.60)

the last inequality being due to just (5.58). By putting v = u+ tw and dividing itby t and assuming that Aε is demicontinuous, one would get Aε:

limt→0

Φε(u+tw)− Φε(u)

t=⟨Aε(u), w

⟩(5.61)

which would show that Φε is Gateaux differentiable.It thus remains to prove the demicontinuity and also the boundedness of Aε.

Taking some u0 ∈ dom(Φ) and f ∈ ∂Φ(u0), testing (5.59) by uε − u0, and using(5.57), i.e. Aε(u) ∈ ∂Φ(uε), and the monotonicity of ∂Φ, we get⟨

J(uε − u), uε − u0

⟩= ε

⟨Aε(u), u0 − uε

⟩≤ ε

(⟨Aε(u), u0 − uε

⟩+⟨f −Aε(u), u0 − uε

⟩)= ε

⟨f, u0 − uε

⟩. (5.62)

Hence

‖uε − u‖2 =⟨J(uε−u), uε−u0

⟩+⟨J(uε−u), u0−u

⟩≤ ε‖f‖∗ ‖u0 − uε‖+ ‖uε − u‖ ‖u− u0‖. (5.63)

This implies that u �→ uε is bounded (i.e. maps bounded sets into bounded sets)and, in view of (5.59), also Aε is bounded.

Now consider uk → u in V and the corresponding ukε := (uk)ε. Again by(5.59), we have J(ukε − uk) + εAε(uk) = 0 and also J(ulε − ul) + εAε(ul) = 0.Subtracting it and testing by ukε − ulε, we obtain

L(1)kl + L

(2)kl :=

⟨J(ukε−uk)− J(ulε−ul), (ukε−uk)− (ulε−ul)

⟩+ ε

⟨Aε(uk)−Aε(ul), ukε−ulε

⟩=⟨J(ukε−uk)− J(ulε−ul), ul−uk

⟩=: Rkl . (5.64)


We have L(1)kl ≥ 0 because J is monotone and also L

(2)kl ≥ 0 because Aε(uk) ∈

∂Φ(ukε) and ∂Φ is monotone. Moreover, limk,l→∞Rkl = 0 because limk,l→∞(uk−ul) = 0 while both J(ukε−uk) and J(ulε−ul) are bounded because the map-

ping u �→ uε has already been shown bounded. Thus limk,l→∞ L(1)kl = 0 and

limk,l→∞ L(2)kl = 0.

Considering (if needed) a subsequence, indexed for simplicity again by k, suchthat ukε ⇀ u in V , Aε(uk) ⇀ f and J(ukε−uk) ⇀ j∗ in V ∗, and 〈Aε(uk), ukε〉 → ξ

in R for k → ∞. From limk,l→∞ L(2)kl = 0 we get

0 = liml→∞

(limk→∞

⟨Aε(uk)−Aε(ul), ukε−ulε

⟩)= lim

l→∞

(ξ − ⟨f, ulε

⟩− ⟨Aε(ul), u−ulε

⟩)= 2ξ − 2

⟨f, u

⟩. (5.65)

Hence 〈Aε(uk), ukε〉 → 〈f, u〉. Since Φ is monotone and Aε(uk) ∈ Φ(ukε), wehave 0 ≤ 〈Aε(uk) − y, ukε − z〉 → 〈f − y, u − z〉. As it holds for any (y, z) suchthat y ∈ ∂Φ(z) and as ∂Φ is maximal monotone, cf. Theorem 5.3(ii), we havef ∈ ∂Φ(u).

Furthermore, from Aε(uk) = J(uk − ukε)/ε and from the definition (3.1) ofJ , we obtain

〈εAε(uk), ukε − uk〉 = ‖ukε − uk‖2 = ε2‖Aε(uk)‖∗. (5.66)

In the limit, ‖u− u‖2 ≤ lim infk→∞ ‖ukε − uk‖2 = limk→∞〈εAε(uk), ukε − uk〉 =〈εf, u−u〉 ≤ ε‖f‖∗‖u−u‖. Hence, in particular, ‖u−u‖ ≤ ε‖f‖∗. Conversely, againby using (5.66), ε‖f‖2∗ ≤ ε lim infk→∞ ‖Aε(uk)‖2∗ = limk→∞〈Aε(uk), uk − ukε〉 =〈f, u − u〉 ≤ ‖f‖∗‖u − u‖, hence ε‖f‖∗ ≤ ‖u − u‖. Hence altogether we provedε‖f‖∗ = ‖u− u‖ and thus also

ε2‖f‖2∗ = ‖u− u‖2 ≤ 〈εf, u− u〉 ≤ ε‖f‖∗‖u− u‖ = ε2‖f‖2∗ (5.67)

hence 〈εf, u− u〉 = ‖f‖2∗. Altogether, we proved J(u − u) + εf = 0.Therefore, u = uε and f = Aε(u). Since this limit is identified uniquely, we

have Aε(uk) ⇀ f = Aε(u) for the whole sequence. �

Remark 5.18 (Yosida approximation of monotone mappings). The differential Φ′εcan be understood as the so-called Yosida approximation of ∂Φ. In general, amonotone mapping Aε : V → V ∗ defined by13

Aε(u) =J(u− (I+εJ−1A)−1(u)

)ε

, (5.68)

is called the Yosida approximation of the monotone (generally non-potential set-valued) mapping A : V ⇒ V ∗; for V = Rn see also (2.164b). Let us note that in the

13Note that J is single-valued as V ∗ is supposed strictly convex; cf. Lemma 3.2(ii).


proof of Lemma 5.17, we actually proved that, if V is strictly convex together withV ∗ and if A is maximal monotone, then Aε : V → V ∗ is monotone, bounded, anddemicontinuous. Moreover, it can be proved that w-limε→0 Aε(u) is the elementof A(u) having the minimal norm, and that, if V is a Hilbert space, Aε is evenLipschitz continuous.

Corollary 5.19. Let V be reflexive, Φ : V → R ∪ {+∞} convex, proper, and lowersemicontinuous, A : V → V ∗ be pseudomonotone and

(i) Φ(v) ≥ ‖v‖α for some α > 1, and 〈A(u),u−v〉‖u‖ be bounded from below for some

v∈DomΦ, or

(ii) Φ be bounded from below and A be coercive in the sense:

∃v ∈ Dom(Φ) : lim‖u‖→∞

〈A(u), u− v〉‖u‖ = +∞. (5.69)

Then for any f ∈ V ∗ there is at least one u ∈ V solving (5.37).

Proof. By Asplund’s theorem, V can be renormed (if needed) so that both V andV ∗ are strictly convex. Then, by Lemma 5.17, Φ possesses the smooth regularizingfamily {Φε}ε>0 with the property (5.38) and with a bounded and demicontinuousΦ′ε. Let us verify (5.39):

The case (i): for any 0 < ε ≤ ε0 with ε0 fixed, one gets14

Φε(u) ≥ Φε0(u) = infv∈V

‖u− v‖22ε0

+Φ(v) ≥ infv∈V

(‖u‖ − ‖v‖)22ε0

+ ‖v‖α

≥ [ | · |α]ε0(‖u‖) ≥ rαε0 , where rε0+ αε0r

α−1ε0 = ‖u‖, (5.70)

where [| · |α]ε0 is the Yosida approximation of the scalar function | · |α; thelast estimate follows likewise the second estimate in (5.55) while the equa-tion for rε0 is an analog of (5.57). This relation allows for an estimate rε0 ≥ε0‖u‖min(1,1/(α−1)) − 1/ε0 for some ε0 > 0, hence (5.70) yields a uniform (andsuperlinear) growth at least as ‖u‖min(α,α′). Then, adding it with the assumedestimate 〈A(u), u − v〉/‖u‖ ≥ −C yields (5.39).

The case (ii): Without loss of generality, we can assume Φ ≥ 0. Then Φε ≥ 0,too. Then, adding Φε(u) to the numerator in (5.69) gives (5.39).

Then the assertion follows by Theorem 5.15. �Theorem 5.20 (Monotone case: uniqueness and stability). Let the assump-tion of Corollary 5.19 be valid with A monotone and radially continuous. Then:

(i) If A is strictly monotone or Φ is strictly convex, then the solution u to (5.37)is unique and the mapping f �→ u is demicontinuous.

(ii) If, in addition, A is d-monotone and V uniformly convex, then f �→ u iscontinuous.

(iii) If A is uniformly monotone, then f �→ u is uniformly continuous.

14We used ‖u− v‖ ≥ max(‖u‖ − ‖v‖, ‖v‖ − ‖u‖) so that also ‖u− v‖2 ≥ (‖u‖ − ‖v‖)2.


Proof. Take u1, u2 ∈ V two solutions to (5.37), i.e., for i = 1, 2,

Φ(v) + 〈A(ui), v − ui〉 ≥ Φ(ui) + 〈f, v − ui〉. (5.71)

For i = 1 take v = u2:

Φ(u2) + 〈A(u1), u2 − u1〉 ≥ Φ(u1) + 〈f, u2 − u1〉 . (5.72)

Analogously, for i = 2 take v = u1. Adding the obtained inequalities, we get

〈A(u1)−A(u2), u2 − u1〉 ≥ 0, (5.73)

from which we get u1 = u2 if A is strictly monotone.If A is merely monotone, we can use strict convexity of Φ as follows: we

consider again (5.71) but now for v = 12u1+

12u2, and use monotonicity of A to get

Φ(ui)− Φ(v) + 〈f, v−ui〉 ≤ 〈A(ui), v−ui〉= 〈A(v), v−ui〉+ 〈A(ui)−A(v), v−ui〉 ≤ 〈A(v), v−ui〉

for i = 1, 2. Summing it together yields

Φ(u1) + Φ(u2)− 2Φ(v) ≤ 〈A(v) − f, 2v − u1 − u2〉 = 0 (5.74)

because obviously 2v−u1−u2 = 0. Yet, (5.74) implies u1=u2 if Φ is strictly convex.For the demicontinuity, we use again the Minty trick: take fi → f and ui the

solution corresponding to fi. By the a-priori estimate as in Corollary 5.19, {ui}i∈Nis bounded. Then ui ⇀ u (for a moment, possibly as a subsequence). Then, by themonotonicity of A, by (5.71) written for fi instead of f , and by the weak lowersemicontinuity of Φ, we get

0 ≤ lim supi→∞

〈A(v)−A(ui), v−ui〉

≤ lim supi→∞

(〈A(v), v−ui〉 − Φ(ui) + Φ(v)− 〈fi, v−ui〉

)≤ 〈A(v), v−u〉 − Φ(u) + Φ(v)− 〈f, v−u〉. (5.75)

In particular, u ∈ dom(Φ). Then, for w ∈ dom(Φ), the convex combination vε :=εw + (1 − ε)u belongs to dom(Φ). Similarly as in the proof of Proposition 5.11,using (5.75) with v := vε and realizing that, by the convexity of Φ, we haveΦ(vε)− Φ(u) ≤ ε(Φ(w) − Φ(u)), we obtain

0 ≤ ⟨A(vε), vε−u

⟩+Φ(vε)− Φ(u)− ⟨f, vε−u

⟩≤ ⟨

A(vε), ε(w−u)⟩+ ε(Φ(w)− Φ(u)

)− ⟨f, ε(w−u)⟩. (5.76)

Dividing it by ε > 0, we come to

〈A(vε), w − u〉+Φ(w) − Φ(u) ≥ 〈f, w − u〉. (5.77)


Then, for ε ↘ 0 by using the radial continuity of A, we get that u solves 〈A(u), w−u〉 + Φ(w) − Φ(u) ≥ 〈f, w − u〉, i.e. u solves (5.37). As we proved such u to beunique, even the whole sequence {ui}i∈N converges weakly to u.

The norm (resp. uniform) continuity in the d-monotone (resp. uniform-monotone) case is a simple modification of (5.71)–(5.72) for f = f1,2 so that(5.73) turns into 〈A(u1) − A(u2), u2 − u1〉 ≤ 〈f1 − f2, u2 − u1〉 and then one canproceed as in (2.33) (resp. in (2.34)). �

Remark 5.21. A special case: Φ = δK , K ⊂ V convex, closed. Then (5.37) turnsinto (5.15), i.e. into the problem

Find u ∈ K : ∀v ∈ K : 〈A(u), v − u〉 ≥ 〈f, v − u〉. (5.78)

Remark 5.22 (Another penalty functional). Considering the constraint of the typeu ∈ K, one may be tempted to consider another norm than Lα(Ω) used in (5.25).Inspired by (5.49), one can consider the functional u �→ 1

2ε

(infv∈K

∫Ω|u − v|p +

|∇(u − v)|p)2/p, which however leads to a nonlocal term in the approximatingequation related, in fact, to the formula (5.68) for A = NK . A certain caution isadvisable: e.g. penalization of K = {v ≥ 0 on Ω, v|Γ = 0} by 1

2ε

∫Ω |∇(u−)|2dx is

not suitable because this functional is not convex.

Remark 5.23 (Abstract Galerkin approximation of variational inequalities15). Wecan adapt the finite-dimensional approximation from Section 2.1. Considering(5.78), instead of uk ∈ Vk solving I∗k (A(uk) − f) = 0, we will now start withuk ∈ Kk ⊂ K solving uk = Pk(uk + J−1

k I∗k (f −A(uk))) where Ik : Vk → V is theinclusion, Jk : Vk → V ∗k is the duality mapping, Pk : Vk → Kk is the projectorwith respect to the Euclidean inner product in Vk (which is thus considered aspossibly renormed) and Kk ⊂ Vk is a convex closed approximation of K whoseunion is dense in K, cf. also Exercise 5.42 below. In other words, uk ∈ Kk satisfies

∀v ∈ Kk :⟨A(uk), v − uk

⟩ ≥ 〈f, v − uk〉. (5.79)

The existence of uk again follows by the Brouwer fixed-point Theorem 1.10. Thusone can show that, if A is pseudomonotone and coercive on K in the sense

∃v ∈ K : lim‖u‖→∞

⟨A(u), u− v

⟩‖u‖ = +∞, (5.80)

cf. (5.69), then, for any f ∈ V ∗, (5.78) has a solution. Actually, this is an alternativeproof of Corollary 5.19 in the special case Φ = δK .

Remark 5.24 (Epigraphical approach). In fact, (5.78) is a universal form for (5.8)if one makes the so-called Mosco transformation [292]: replace V by V × R, putK :=epi(Φ1) ⊂ V ×R, define the pseudomonotone mapping A : V ×R → V ∗ ×R

15See Brezis [65] or Lions [261, Sect. II.8.2].


by A(u, a) := (A2(u), 1), and the right-hand side (f, 0). Indeed, if (u, a) ∈ K solvesthe problem (5.78) for such data, i.e. if Φ1(u) ≤ a and, for all (v, b) ∈ V × R,

Φ1(v) ≤ b ⇒ ⟨(A2(u), 1) , (v, b)− (u, a)

⟩ ≥ ⟨(f, 0) , (v, b)− (u, a)⟩, (5.81)

then a = Φ1(u) and u solves (5.8).16 The previous Remark 5.23 allows us to givean alternative proof of Corollary 5.19 under the following coercivity condition:

∃v ∈ dom(Φ1) : lim‖u‖→∞

Φ1(u) + 〈A2(u), u− v〉‖u‖ = +∞, (5.82)

which covers (and generalizes) both cases (i) and (ii) in Corollary 5.19.

5.4 Excursion to quasivariational inequalities

There is a sensible generalization of (5.8) by allowing the convex functional Φ1 todepend on the solution u itself, i.e. Φ1 = Φ(u, ·) for some Φ : V ×V → R. For A =A2 monotone (not necessarily potential) we come to a so-called quasivariationalinequality

∀v ∈ V : Φ(u, v) + 〈A(u), v − u〉 ≥ Φ(u, u) + 〈f, v − u〉. (5.83)

To prove the existence of a solution to (5.83), various fixed-point theoremsare usually used. Here, we use the Kakutani’s Theorem 1.11. We denote by M(w)the set of the solutions u to the following auxiliary variational inequality:

∀v ∈ V : Φ(w, v) + 〈A(u), v − u〉 ≥ Φ(w, u) + 〈f, v − u〉. (5.84)

Lemma 5.25. Let A be monotone, bounded, radially continuous, w �→ Φ(w, ·) beweakly continuous in Mosco’s sense, i.e. for all v, w ∈ V ,

∀wk ⇀ w ∃vk → v : lim supwk⇀wvk→v

Φ(wk, vk) ≤ Φ(w, v), (5.85a)

∀wk ⇀ w ∀vk ⇀ v : lim infwk⇀wvk⇀v

Φ(wk, vk) ≥ Φ(w, v), (5.85b)

and, for any w ∈ V , Φ(w, ·) ≥ 0 be convex, dom(Φ(w, ·)) � 0, and A be coercive.Then M(w) := {u ∈ V ; u solves (5.84)} is nonempty, closed and convex, andM : V ⇒ V is weakly upper semicontinuous, i.e.17

wk ⇀ w,uk ⇀ u,

uk∈M(wk)

⎫⎬⎭ ⇒ u ∈ M(w). (5.86)

16Indeed, choosing (v, b) := (u,Φ1(u)) in (5.81), we get Φ1(u) ≥ a, hence Φ1(u) = a. By thisand by putting (v, b) := (v,Φ1(v)) into (5.81), we get just (5.8) with v arbitrary.

17Let us recall the generally applied “sequential” concept, i.e. (5.86) defines “sequential” weakupper semicontinuity. This is because the generally assumed separability and reflexivity of V(hence of V ∗ too) makes the weak topology metrizable if restricted to bounded sets and thenthe “sequential” concept can be applied equally as the usual general-topology concept.

5.4. Excursion to quasivariational inequalities 155

Proof. By (5.85b), in particular, Φ(w, ·) is weakly lower semicontinuous and then,by pseudomonotonicity of A and the coercivity, (5.84) has a solution; cf. Corol-lary 5.19. Hence M(w) �= ∅.

Take u1 and u2 two solutions to (5.84), i.e. after a trivial re-arrangement:

Φ(w, v) −Φ(w, u1) + 〈f, u1 − v〉 ≥ 〈A(u1), u1 − v〉, (5.87a)

Φ(w, v) −Φ(w, u2) + 〈f, u2 − v〉 ≥ 〈A(u2), u2 − v〉. (5.87b)

Then we add it together, divide it by 2, and subtract the trivial identity 〈A(v), u−v〉 = 1

2 〈A(v), u1 − v〉+ 12 〈A(v), u2 − v〉 where u = 1

2u1 +12u2. Using subsequently

the convexity of Φ(w, ·), (5.87), and the monotonicity of A, we get

Φ(w, v) −Φ(w, u) +⟨f −A(v), u − v

⟩ ≥ Φ(w, v)− 1

2Φ(w, u1)− 1

2Φ(w, u2)

+⟨f,

1

2u1 +

1

2u2 − v

⟩− 1

2

⟨A(v), u1 − v

⟩− 1

2

⟨A(v), u2 − v

⟩≥ 1

2

⟨A(u1)−A(v), u1 − v

⟩+

1

2

⟨A(u2)−A(v), u2 − v

⟩ ≥ 0.

This is essentially the desired inequality if one replaces A(v) by A(u), which canbe however made by Minty’s trick by putting v = εz + (1 − ε)u with 0 < ε ≤ 1and proceed as in (5.76)–(5.77). This shows that u ∈ M(w). As u1 and u2 werearbitrary, by Proposition 1.6, M(w) is shown convex if closed. This closednessfollows from (5.86). To show it, take wk ⇀ w and uk ⇀ u such that uk ∈ M(wk).In view of (5.84) for wk instead of w, this means for any vk ∈ V :

Φ(wk, vk) + 〈A(uk), vk − uk〉 ≥ Φ(wk, uk) + 〈f, vk − uk〉. (5.88)

Now we consider v ∈ V arbitrary and a suitable sequence {vk}k∈N converging tov so that (5.85a) holds. By Lemma 2.9, A is pseudomonotone, and thus we canpass to the limit in (5.88) entirely similarly as in the proof of Theorem 5.15 withRemark 5.16, which gives (5.84).18 Hence u ∈ M(w), as claimed in (5.86). Inparticular, for wk ≡ w we get that M(w) is closed. �

Theorem 5.26. Let the assumptions of Lemma 5.25 be fulfilled with Φ(w, 0) ≤C(1 + ‖w‖) with C < +∞. Then (5.83) has a solution.

Proof. By (5.84) with v = 0 and by the assumed coercivity of A, for any u ∈ M(w),w ∈ V , we have the a-priori estimate:

ζ(‖u‖)‖u‖ ≤ 〈A(u), u〉 ≤ Φ(w, u) + 〈A(u), u〉≤ Φ(w, 0) + 〈f, u〉 ≤ C

(1 + ‖w‖)+ ‖f‖∗‖u‖

18To make this limit passage more direct without using the pseudomonotone argument, oneneeds additionally continuity of A, cf. Exercise 5.38 below.


with some ζ : R+ → R+ such that limr→∞ ζ(r) = +∞. Divided by ‖u‖, this gives

ζ(‖u‖) ≤ C1 + ‖w‖‖u‖ + ‖f‖∗ (5.89)

from which we can see that M(B) ⊂ B for a sufficiently large ball B ⊂ V .

By Lemma 5.25, we can thus use the Kakutani fixed-point Theorem 1.11 forthe ball B endowed with a weak topology to show the existence of u ∈ V suchthat M(u) � u. Such u obviously solves (5.83). �

Example 5.27. For the typical case Φ(w, u) = δK(w)(u), (5.83) turns into thequasivariational inequality:

Find u∈K(u) : ∀v∈K(u) : 〈A(u), v − u〉 ≥ 〈f, v − u〉. (5.90)

Then (5.85a) means that the set-valued mapping K : V ⇒ V is so-called(weak,norm)-lower semicontinuous in the Kuratowski sense19 while (5.85b) is just(weak,weak)-upper semicontinuity20.

Example 5.28. Let us consider V = W 1,20 (Ω), A = −Δ, and

Φ(w, v) :=

∫Ω

ϕ(x,w(x), v(x)) dx (5.91)

with ϕ : Ω×R×R → R a Caratheodory mapping satisfying the growth condition

∃a∈L1(Ω), b∈R+ : |ϕ(x, r1, r2)| ≤ a(x) + b(|r1|2∗−ε + |r2|2∗−ε

). (5.92)

Then Φ : W 1,2(Ω) ×W 1,2(Ω) → R is (weak×weak)-continuous; use the compactembeddingW 1,2(Ω) � L2∗−ε(Ω) and then the continuity of the Nemytskiı mappingNϕ : L2∗−ε(Ω)× L2∗−ε(Ω) → L1(Ω).

Supposing additionally that ϕ(x, r1, ·) ≥ 0 is convex and ϕ(x, r1, 0) ≤ γ(x)+C|r1| with some γ ∈ L1(Ω) and C ∈ R, we can prove the existence of a solutionu∈W 1,2

0 (Ω) to the quasivariational inequality∫Ω

ϕ(x, u, v) +∇u · ∇(v − u) dx ≥∫Ω

ϕ(x, u, u) + g(v − u) dx (5.93)

for any v∈W 1,20 (Ω) which corresponds, in the classical formulation, to the problem:

−Δu+ ∂r2ϕ(u, u) � g in Ω,

u = 0 on Γ.

}(5.94)

19This is, by definition: ∀uk ⇀ u ∈ V ∀v ∈ K(u) ∃vk ∈ K(uk): vk → v.20This is, by definition, just (5.86) with K in place of M .

5.5. Exercises 157

5.5 Exercises

Exercise 5.29. Specify the potential Φ of A from Figure 10b and c.21

Exercise 5.30. By using Proposition 1.6, show that any convex lower semicontin-uous functional Φ : V → R ∪ {+∞} is weakly lower semicontinuous.22

Exercise 5.31. Show ∂(Φ1+Φ2)(u) ⊂ ∂Φ1(u)+∂Φ2(u) for Φ1,Φ2 : V→R convex.23

Exercise 5.32. Show that Φ convex and ∂Φ : V ⇒ V ∗ coercive imply Φ coercive.24

Exercise 5.33. Assuming Φ convex and lower semicontinuous, prove that the graphof the multivalued mapping ∂Φ : V ⇒ V ∗ is (weak×norm)- and (norm×weak)-closed.25

Exercise 5.34. Show that, if Φ : V → R is Gateaux differentiable and convex, then∂Φ(u) = {Φ′(u)}.26

Exercise 5.35. Modify the proof of Theorem 5.15 if Φε → Φ only in Mosco’s sense,i.e. (5.38b)–(5.48).27

Exercise 5.36. Modify Theorem 5.20(i) for the case A = A1+A2 with A1 monotoneand radially continuous and A2 totally continuous, to obtain upper semicontinuityof the set-valued mapping f �→ {u ∈ V ; u solves (5.37)} as (V ∗,norm)⇒ (V,weak).If A2 = 0, show that this set-valued mapping has convex values.28

Exercise 5.37. Verify the convergence Φε → Φ in the sense (5.38) for Φ = δK andΦε(u) =

12εdist(u,K)2 directly, without using Lemma 5.17.

Exercise 5.38. Strengthening the assumptions in Lemma 5.25 by requiring A notonly demicontinuous (as Lemma 2.16 says) but even continuous, perform the limit

21Hint: The absolute value | · | and the indicator function δ[−1,1](·), up to a constant, of course.22Hint: Assume the contrary, i.e. l := limk→∞ Φ(uk) < Φ(u) for some uk ⇀ u, and realize

that the level set L = {v ∈ V ; Φ(v) ≤ l} is convex and closed because Φ is convex andlower semicontinuous. By Proposition 1.6, L is weakly closed, so that L �w-limk→∞ uk = u,i.e. Φ(u) ≤ l, a contradiction.

23Hint: It follows directly from the definition (5.2).24Hint: Modify the proof of Theorem 4.4(i).25Hint: Assume either uk ⇀ u and fk → f or uk → u and fk ⇀ f , and make a limit passage

in the inequality in (5.2).26Hint: By (5.10), 〈Φ′(u), v − u〉 = DΦ(u, v − u) ≤ Φ(v) − Φ(u), hence Φ′(u) ∈ ∂Φ(u).

Conversely, consider f ∈ ∂Φ(u), i.e. Φ(v) − Φ(u) ≥ 〈f, v − u〉 for all v, and in particular forv := u+ εw, hence (Φ(u + εw)− Φ(u))/ε ≥ 〈f, w〉. For ε ↘ 0, deduce DΦ(u, w) ≥ 〈f, w〉. SinceDΦ(u,w) = 〈Φ′(u), w〉, hence f=Φ′(u).

27Hint: Put vε instead of v into (5.41) to be used for (5.43), and then use lim supε→0〈A(uε), vε−uε〉 = limε→0〈A(uε), vε − v〉+ lim supε→0〈A(uε), v− uε〉 where the first right-hand-side term iszero if vε → v, as assumed in (5.48).

28Hint: For two solutions u1 and u2, write 2⟨f−A(v), u−v⟩ =

⟨f−A(v), u1−v

⟩+⟨f−A(v), u2−

v⟩ ≥ ⟨

A(u1)−A(v), u1−v⟩+Φ(u1)−Φ(v)+

⟨A(u2)−A(v), u2−v

⟩+Φ(u2)−Φ(v) ≥ 2Φ(u)−2Φ(v),

cf. also (2.28) for Φ = 0, and the use the Minty trick as in the proof of Theorem 5.20(i).


passage in (5.88) by Minty’s trick.29

Exercise 5.39 (Two-sided obstacles). Consider w1, w2 ∈ W 1,p(Ω), w1 < w2 inΩ, K = {v ∈ W 1,p(Ω); w1 ≤ v ≤ w2}, and the variational inequality (5.19).Formulate the complementarity problem like (5.18) for this case and modify theproof of Proposition 5.9 accordingly.

Exercise 5.40 (Obstacle on Γ). For some w ∈ W 1−1/p,p(Γ), consider the so-calledSignorini-type problem, i.e. (in the classical formulation) a problem involving thecomplementarity Signorini-type boundary conditions on Γ only:


)+ |u|q−2u = g in Ω,

|∇u|p−2 ∂u∂ν + b(x, u) ≥ h,

u ≥ w,(|∇u|p−2 ∂u∂ν + b(x, u)− h

)(u− w) = 0

⎫⎪⎬⎪⎭ on Γ

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(5.95)

with b qualified as in (5.26b) and 1 < q ≤ p∗. Assemble the weak formulationwhich involves the convex set

K :={u∈W 1,p(Ω); u(x) ≥ w(x) for a.a. x∈Γ

}(5.96)

and show the relation with the classical formulation.30 Modify (5.25): consider q ≤p# and the penalty term in the form (εα)−1

∫Γ |(w−u)+|α dS. Show the a-priori

estimates ‖uε‖W 1,p(Ω) = O(1) and ‖(w−uε)+‖Lα(Γ) = O( α

√ε) and the convergence

for ε → 0.31 Perform the analysis for q > p∗ by modifing the space V .32 Further,modify the equation by adding �c(u)·∇u or c(∇u) as in Exercise 5.46.

Exercise 5.41 (Dirichlet boundary condition). Instead of (5.96), consider

K :={u∈W 1,p(Ω); u ≥ w on Ω, u|Γ = u

Don Γ

}, (5.97)

which is nonempty if w|Γ ≤ uD. Modify Section 5.2.

Exercise 5.42 (Ritz’ method). Consider Ω polygonal, Φ from (4.23), K ={v ∈

W 1,p(Ω); v ≥ w in Ω}as in (5.20), Vk a finite-dimensional subspace of W 1,p(Ω)

constructed by the piece-wise affine finite elements, cf. Example 2.67. As in (2.61),

29Hint: Write 0 ≤ 〈A(vk) − A(uk), vk − uk〉 ≤ 〈A(vk), vk − uk〉+Φ(wk, vk)−Φ(wk, uk), use(5.85) together with A(vk) → A(v) so that 〈A(vk), vk − uk〉 → 〈A(v), v − u〉 and finish by theMinty trick as in the proof of Theorem 5.20(i).

30Hint: Just simplify the proof of Proposition 5.9.31Hint: Use the test by v = uε − w with a suitable extension w ∈ W 1,p(Ω) of the originally

given w ∈ W 1−1/p,p(Γ). Realize how (5.26a) can be verified, namely |s|p−2s·(s−∇w(x)) ≥|s|p− |s|p−1|∇w(x)| ≥ 1

p|s|p− 1

p|∇w(x)|p and |r|q−2r·(r−w(x)) ≥ |r|q − |r|q−1|w(x)| ≥ 1

q|r|q −

1q|w(x)|q. For convergence, modify the proof of Proposition 5.10.32Hint: Like (2.128), consider V = W 1,p(Ω) ∩ Lq(Ω). Assuming w ∈ V , make estimation like

in Remark 5.12; in particular, estimate∫Ω |uε|q−2uεw dx ≤ 1

q‖uε‖qLq(Ω)

+ 1q‖w‖q

Lq(Ω).

5.5. Exercises 159

define Φ0(u) = Φ(u+w) and prove existence of a minimizer uk ∈ Vk of Φ0 subjectto uk ≥ 0 a.e. in Ω. Further, prove33

cl( ⋃

k∈N

{v∈Vk; v ≥ 0

})={v∈W 1,p(Ω); v ≥ 0

}, (5.98)

derive a-priori estimates, and show convergence by a direct method, i.e. with-out Minty’s trick, as in (5.28)–(5.29), of uk to u0, a minimizer of Φ0 on {v ∈W 1,p(Ω); v ≥ 0

}. Show that u = u0 + w solves the original variational problem,

more precisely it minimizes of Φ on K.

Exercise 5.43 (Galerkin method). Consider A(u) := −div(A∇u) with A ∈ Rn×n

positive definite (but, in general, nonsymmetric hence the problem is non-potential) on a polygonal domain Ω, and the unilateral problem

−div(A∇u) ≥ gu ≥ w(

div(A∇u) + g)(u − w) = 0

⎫⎬⎭ on Ω,

u = 0 on Γ.

⎫⎪⎪⎬⎪⎪⎭ (5.99)

Use the transformation (2.61) to get a problem like (5.99) but with g+div(A∇w)and 0 in place of g and w, respectively. Make the approximation by a finite-dimensional subspace Vk of W 1,p(Ω), use (5.98), derive a-priori estimates andshow convergence either by Minty’s trick or by a direct limit passage.34

Exercise 5.44 (Regularization of the elliptic variational inequality I ). Con-sider (5.18) without symmetry (4.21), thus without any potential, and the regular-ization (5.25). Assume a(x, r, ·) monotone and b = b(r) and c = c(r) having a sub-critical growth, i.e. (2.56b,c) holds. Show a-priori estimates ‖uε‖W 1,p(Ω) = O(1)and ‖(w − uε)

+‖L2(Ω) = O(√ε).35 Further, show the convergence uε ⇀ u in

W 1,p(Ω) with u being a weak solution to (5.18) by using the Minty trick.36

33Hint: Realize density of smooth non-negative functions in {v∈W 1,p(Ω); v ≥ 0}, which canbe proved by applying a convolution with a mollifier (for n = 1 see also (7.11) in Sect. 7.1 andFigure 16). Note that the general constraint v ≥ w would not be preserved by mollifying v, whichis why the shift Φ0(u) = Φ(u+w) was made.

34Hint: Use lim supk→∞∫Ω(∇v −∇uk)

A∇uk dx ≤ ∫Ω(∇v −∇u0)A∇u0 dx if uk → u0.

35Hint: This is essentially as in the proof of Proposition 5.10.36Hint: modify the proof of Proposition 5.11. By a(x, r, ·), instead of (5.33), arrive to

0 ≤∫Ω(a(uε,∇uε)−a(uε,∇v))·∇(uε−v) dx

≤∫Ω

(g − c(uε)

)(uε−v) − a(uε,∇v)·∇(uε−v) dx+

∫Γ

(h− b(uε)

)(uε−v) dS

→∫Ω

(g−c(u))(u−v) − a(u,∇v)·∇(u−v) dx+

∫Γ

(h− b(u)

)(u−v) dS,

where the limit passage in lower-order terms used c(uε) → c(u) in Lp∗′(Ω) and a(uε,∇v) →

a(u,∇v) in Lp′(Ω;Rn) by compactness of the embedding W 1,p(Ω) ⊂ Lp∗−ε(Ω) and continuity

of the Nemytskiı mappings Nc and Na(·,·,∇v), and also b(uε)→ b(u) in Lp#′Γ) by compactness

of the trace operator W 1,p(Ω)→ Lp#−ε(Γ) and continuity of the Nemytskiı mapping Nb.


Exercise 5.45. Modify the approach from Exercise 5.44 by starting directly withthe monotonicity of

(r, s) �→(− 1

ε

((w(x)−r)+)α−1, a(x, r, s

)): R×Rn → R×Rn (5.100)

for any r fixed instead of the monotonicity of a(x, r, ·) : Rn → Rn.

Exercise 5.46 (Regularization of the elliptic variational inequality II). Consider

−Δuε + c(∇uε) +1

εu+ε = g in Ω,

uε = 0 on Γ,

⎫⎬⎭ (5.101)

where u+ = max(0, u), with c continuous of sub-linear growth, i.e. |c(s)| ≤ C(1 +|s|1−ε) as in Exercise 2.86 for p = 2. Show a-priori estimates ‖uε‖W 1,2(Ω) = O(1)and ‖u+

ε ‖L2(Ω) = O(√ε).37 Further show convergence to the weak solution to the

complementarity problem:38

−Δu+ c(∇u) ≤ g ,

u ≤ 0,(Δu− c(∇u) + g

)u = 0,

⎫⎪⎬⎪⎭ in Ω,

u = 0 on Γ.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (5.102)

Exercise 5.47 (Bingham-fluid-like model). Consider the potential:39

Φ(u) =

∫Ω

ε|∇u|2 + |∇u|+ δ|u− u|2 + fu dx (5.103)

with ε > 0 a regularization parameter, u∈L2(Ω) given. Show the existence of aunique minimizer u∈W 1,2

0 (Ω). Formulate the corresponding variational inequality.

37Hint: Test (5.101) by uε.38Hint: By the a-priori estimates we can select a subsequence uε ⇀ u in W 1,2(Ω), u ≤ 0 a.e. in

Ω. For any v ∈W 1,2(Ω), v ≤ 0, by using (5.101),∫Ω|∇uε −∇v|2dx ≤

∫Ω|∇uε −∇v|2 +

1

ε(u+

ε − v+)(uε − v) dx

=

∫Ω(g − c(∇uε))(uε − v) −∇v·∇(uε − v) − 1

εv+(uε − v) dx.

Note that the last term vanishes as v ≤ 0. In particular, take v := u to see that uε → u inW 1,2

0 (Ω). Thus pass to the limit in the nonlinear Nemytskiı mapping Nc, i.e. c(∇uε)→ c(∇u).Then, by (5.101) tested by v − uε, make a limit passage in∫

Ω∇uε · ∇(v − uε) + (c(∇uε)− g)(v − uε)dx = −1

ε

∫Ωu+ε (v − uε) dx ≥ 0,

provided v ≤ 0, which gives the weak formulation of (5.102).39For δ = 0, this is a scalar version of a so-called Bingham-fluid model, while in case n = 2

and f = 0 this problem has another interpretation in image enhancement/reconstruction.

5.5. Exercises 161

Exercise 5.48 (Plasticity-like model40). Consider the problem: find u ∈ W 1,∞0 (Ω)

such that |∇u| ≤ 1 a.e. in Ω and

∀v∈W 1,∞0 (Ω), |∇v| ≤ 1 (a.e.) :

∫Ω

a(∇u)·∇(v−u)dx ≥∫Ω

g(v−u) dx. (5.104)

Moreover, assume a(s) · s ≥ |s|p and |a(s)| ≤ C(1 + |s|p−1), and a(·) monotone,

and use a penalization by the functional u �→ ∫Ω

((|∇u|−1)+

)2dx/(2ε). Show that

it leads to the approximate problem (in the classical formulation):41

−div(a(∇uε) +

(|∇uε| − 1)+

ε|∇uε| ∇uε

)= g in Ω,

uε = 0 on Γ.

⎫⎪⎬⎪⎭ (5.105)

Further, assume g ∈ Lmax(2∗,p∗)′(Ω), and show existence of a weak solution

uε ∈ W1,max(2,p)0 (Ω) to (5.105), a-priori estimates by testing (5.105) by uε

42 and

convergence uε ⇀ u in W1,max(2,p)0 (Ω) where u ∈ W 1,∞

0 (Ω) satisfies (5.104)43.Eventually, modify the whole procedure for a ≡ 0.

Exercise 5.49 (Quasivariational inequality). Verify (5.85) for the case Φ(w, u) =δK(w)(u) provided K(w) := {v∈W 1,p

0 (Ω); |∇v(x)| ≤ m(w(x)) for a.a. x∈Ω} withp > n and m : R → R+ continuous, m(·) ≥ ε > 0.44 Assuming a(s) · s ≥ |s|p and

40This complementarity problem is related to a stress field in an elastic/plastic (or, rather,inelastic) bar undergoing a torsion via a Haar-Karman principle; n = 2 and Ω ⊂ R2 is thenthe cross-section. See e.g. Elliott and Ockendon [135, Sect.IV.6], Friedman [155], or Glowinskiet al. [182, p.6 & Chap.3]. Alternatively, the variant a ≡ 0 is related to (a steady-state of) asand flow; in the evolution variant see Aronsson, Evans, Wu [19]. For the classical formulationof (5.104) see (5.108) below.

41Hint: The directional derivative at uε in the direction v is 1ε

∫Ω(|∇uε|−1)+ ∇uε

|∇uε| · ∇v dx.42Hint: Test (5.105) by uε and realize the estimate s·s ≥ (|s|−1)+|s| for s∈Rn, so that∫

Ω

(|∇uε| − 1)+

ε|∇uε| ∇uε · ∇uεdx ≥ 1

ε

∫Ω

((|∇uε| − 1)+)2

dx.

43Hint: Test (5.105) by v − uε with |∇v| ≤ 1 a.e. in Ω, realize that

(|∇uε| − 1)+

ε|∇uε|∇uε · (∇v −∇uε) ≤ 0 a.e. on Ω

and continue the proof as in (5.32) by showing the strong convergence in W 1,p(Ω). To show that|∇u| ≤ 1 a.e. in Ω, estimate the limit inferior in the estimate∥∥(|∇uε| − 1)+

∥∥L2(Ω)

= O(√ε).

44Hint: For (5.85a), one needs to show: ∀wk ⇀ w in W 1,p0 (Ω) ∀u ∈ W 1,p

0 (Ω), |∇u| ≤ m(w)

∃uk: uk → u in W 1,p0 (Ω) and |∇uk| ≤ m(wk) for all k ∈ N. By the compact embedding

W 1,p0 (Ω) � L∞(Ω), realize that m(wk) → m(w) in L∞(Ω), take uk := λku with λk :=

minΩ(m(wk)/m(w)

) → 1. For (5.85b), one needs to pass to the limit in |∇uk| ≤ m(wk) if


|a(s)| ≤ C(1 + |s|p−1), show the existence of a weak solution in W 1,p(Ω). Notethat, for m = 1, one arrives at Exercise 5.48.

Exercise 5.50. Modify Example 5.28 for Φ(w, v) :=∫Ωϕ(x,w(x),∇v(x)) dx with

ϕ = ϕ(x, r, s). Thus solve the inclusion −Δu − div(∂sϕ(u,∇u)) � g with theboundary condition ∂

∂νu+ ν · ∂sϕ(u,∇u)) � 0, which modifies (5.94).

Exercise 5.51 (Dual problem). Consider Exercise 5.41 with uD = 0 and the p-Laplacean, i.e. the complementarity problem


) ≥ g, u ≥ w ,(div(|∇u|p−2∇u

)+ g

)(u−w

)= 0

}in Ω,

u = 0 on Γ,

⎫⎪⎬⎪⎭ (5.106)

and, using (8.261) on p.299, show that the dual problem uses the convex functionalΨ from (5.36) in the form45

Ψ(λ) =

∫Ω

wλ − 1

p′∣∣∇Δ−1

p (g−λ)∣∣pdx, λ∈W 1,p

0 (Ω)∗ ∼= W−1,p′(Ω). (5.107)

Exercise 5.52 (Plasticity-like model II). Consider the potential

Φ(u) :=

∫Ω

ϕ(u,∇u)− gu+ δK(·)(∇u) dx with K(x) ={s∈Rn; |s| ≤ m(x)

}on W 1,p

0 (Ω). Realizing that δK(x)(s) = supλ∈Rn s·λ−m(x)|λ| and defining, like inRemark 5.14, the Lagrangean by L(u, λ) :=

∫Ω ϕ(u,∇u) − gu + λ·∇u − m|λ| dx,

identify the underlying variational inequality as the classical formulation of theconditions for (u, λ) to be a critical point of L,46 i.e. the conditions L′u(u, λ) = 0and ∂λL(u, λ) � 0, and show, in particular, that the classical formulation of (5.104)looks like47

−div(a(∇u) + λ

)= g,

λ ∈ NB1(∇u),

|∇u| ≤ 1

⎫⎪⎬⎪⎭ in Ω,

u = 0 on Γ,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (5.108)

with B1 denoting here the unit ball in Rn.

(uk, wk) ⇀ (u, w) in W 1,p0 (Ω)2. Again, by W 1,p

0 (Ω) � L∞(Ω), m(wk)→ m(w) in L∞(Ω). More-over, for every M ⊂ Ω measurable, by weak lower semicontinuity of convex continuous functions,∫M m(wk) dx = limk→∞

∫M m(wk) dx ≥ lim infk→∞

∫M |∇uk|dx ≥ ∫

M |∇u|dx, from whichm(w) ≥ |∇u| a.e. in Ω.

45Hint: Read (8.261) as minu∈W

1,p0 (Ω)

∫Ω

1p|∇u|p−ξudx=− sup

u∈W1,p0 (Ω)

∫Ω ξu−1

p|∇u|pdx=

− 1p′ ‖∇Δ−1

p ξ‖pLp(Ω;Rn)

and then substitute ξ := g − λ.46Hint: L′

u(u, λ) = 0 means div(a(u,∇u)+λ

)−c(u,∇u)+g = 0 with a and c from (4.23) while∂λL(u, λ) � 0 has a local character m|s|−∇u(x)·(s−λ(x)) ≥ m|λ(x)| for all s∈Rn and a.a. x∈Ω,which equivalently means λ ∈ Bm(∇u) a.e. with Bm being the ball in Rn of the radius m.

47Hint: use just a special data m = 1 and ϕ = ϕ(∇u) so that a = ϕ′.

5.6. Some applications to free-boundary problems 163

5.6 Some applications to free-boundary problems

Variational inequalities are often directly fitted with various unilateral problemsnaturally arising in sciences, as the unilateral contact problem on Figure 12. Some-times, (quasi)variational inequalities arise from concrete free-boundary problemsonly after sophisticated transformations, which is illustrated in this section inconcrete cases.

5.6.1 Porous media flow: a potential variational inequality

We consider the simplest model of a porous, permeable, isotropic, and homoge-neous medium undergoing a flow (a seepage) of an incompressible fluid in a wet,fully saturated domain while the rest is completely dry. Another simplificationconcerns a geometry consisting in a cylindrical vertically oriented domain; to bemore specific, let us consider two reservoirs adjacent to this domain which canbe then considered as a dam. In addition, we consider nonpermeability of the flathorizontal support and of the sides which are not adjacent to any reservoir, andno source on the free boundary (i.e. no contribution by rain water).48 We use thenotation (cf. Fig. 13 on p. 165):

v velocity of the flow,π a piesometric head; we consider49 π = x3 + p, where p is a pressure,ϕ : R2 → R a function whose graph is the free boundary x3 = ϕ(x1, x2),h

Uthe altitude of the upper reservoir,

hL the altitude of the lower reservoir,k the permeability coefficient.

The seepage flow is then governed by Darcy’s law together, of course, with thecontinuity equation, i.e. respectively50

v = −k∇π , (5.109a)

div v = 0 . (5.109b)

This gives kΔπ = 0 so that

Δp = Δ(π − x3) = Δπ = 0. (5.110)

On the free boundary, whose position is not known a-priori, there are two condi-tions

p = 0 and v · ν = 0, (5.111)

48See the monographs by Baiocchi and Capelo [30, Chapter 8], Chipot [99, Chapter 4], Crank[111, Chapter 2], Duvaut and Lions [130, Appendix 2], Elliott and Ockendon [135, Sect. IV.4],Friedman [155], Rodrigues [354, Sect. 2.3], where more general situations can be found, too.

49More generally, one should consider π = x3 + p/(�g) with � the mass density and g gravityacceleration. Here we put �g = 1 for simplicity.

50Note that (5.109) arises from the so-called Darcy-Brinkman system �(v·∇)v− μΔv + v/k+∇π = 0 and div v = 0 when viscosity μ and inertia by mass density � are neglected. This systemmodifies the Navier-Stokes equation, see (6.49) below, for the flow in porous media where kdepends on porosity. Cf. also (8.208) on p. 281 for the evolution variant.


which would seemingly create an overdetermination if it were not the fact that theposition of the free boundary itself is not determined in advance. In (5.111), ν isthe unit normal to the free boundary oriented from the dry region to the wet one,which, in terms of ϕ, means

ν =

(∂ϕ∂x1

, ∂ϕ∂x2

,−1)

√∣∣ ∂ϕ∂x1

∣∣2 + ∣∣ ∂ϕ∂x2

∣∣2 + 1. (5.112)

Comparing (5.109a) with the second condition in (5.111) yields ∂∂ν p = − ∂

∂νx3 =−ν3, so that (5.112) then results in

∂p

∂x1

∂ϕ

∂x1+

∂p

∂x2

∂ϕ

∂x2− ∂p

∂x3= 1. (5.113)

The other boundary conditions are outlined in the left-hand part of Figure 13.We apply the so-called Baiocchi transformation:

u(x) ≡ u(x1, x2, x3) :=

⎧⎨⎩∫ ϕ(x1,x2)

x3

p(x1, x2, ξ) dξ for x3 ≤ ϕ(x1, x2),

0 for x3 > ϕ(x1, x3).

(5.114)

Obviously, ∂∂x3

u = −p. In view of (5.110), we get:

Δu = g(x) on Ω+ :={x∈Ω; u(x) > 0

}; (5.115)

where we implicitly assume p > 0 so that Ω+ represents the wet region. To deter-mine g, let us apply ∂

∂x1and ∂

∂x2to (5.114), which gives

∂u

∂xi=

∫ ϕ(x1,x2)

x3

∂p(x1, x2, ξ)

∂xidξ

+∂ϕ

∂xip(x1, x2, ϕ(x1, x2)

)=

∫ ϕ(x1,x2)

x3

∂p(x1, x2, ξ)

∂xidξ (5.116)

for i = 1, 2 because p(x1, x2, ϕ(x1, x2)) = 0. Applying again ∂∂xi

and using both(5.110) and (5.113), we obtain

∂2u

∂x21

+∂2u

∂x22

=

∫ ϕ(x1,x2)

x3

∂2p(x1, x2, ξ)

∂x21

+∂2p(x1, x2, ξ)

∂x22

dξ

+∂ϕ

∂x1

∂p

∂x1

(x1, x2, ϕ(x1, x2)

)+

∂ϕ

∂x2

∂p

∂x1

(x1, x2, ϕ(x1, x2)

)=

∫ ϕ(x1,x2)

x3

−∂2p(x1, x2, ξ)

∂x23

dξ + 1 +∂p

∂x3

(x1, x2, ϕ(x1, x2)

)=[− ∂p

∂x3(x1, x2, ξ)

]ϕ(x1,x2)

ξ=x3

+ 1 +∂p

∂x3

(x1, x2, ϕ(x1, x2)

)= 1 +

∂p

∂x3= 1− ∂2u

∂x23

. (5.117)


Comparing it with (5.115), we get g = 1.

Ω

x1x2

x3

u=0

u=0

u=0p=0

p=0

u=wv·ν=0

v·ν=0

p=hL−x3 u= 12(hL−x3)

2

u= 12(hU−x3)

2

p=hU−x3

hU

hL

lower

upperreservoir

reservoir

nonpermeable sides

free boundarydry region

wet region(saturated)

nonpermeable bottom

Figure 13. Geometric configuration of the dam problem and boundary conditions;original (left) and transformed (right).

The boundary conditions on the vertical sides are either the Dirichlet or theNeumann ones:51

u =

⎧⎪⎪⎪⎨⎪⎪⎪⎩0 on the upper side,12

((h

U−x3)

+)2

on the side adjacent to the upper reservoir,12

((hL−x3)

+)2

on the side adjacent to the lower reservoir,

w on the bottom, nonpermeable side,

(5.118a)

∂u

∂ν= 0 on the vertical nonpermeable sides. (5.118b)

The Dirichlet boundary condition at the bottom part uses continuity of u

and ∂2

∂x21u + ∂2

∂x22u = 0,52 which implies that the function w = w(x1, x2) occurring

in (5.118) can be determined as the unique solution to the following 2-dimensionalboundary-value problem:

∂2w

∂x21

+∂2w

∂x22

= 0 on the bottom side,

w = 12h

2U

on the bottom edge adjacent to the upper reservoir,

w = 12h

2L

on the bottom edge adjacent to the lower reservoir,∂w

∂ν= 0 on the bottom edges adjacent

to the nonpermeable sides.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭(5.119)

These boundary conditions are outlined in the right-hand part of Figure 13.

51On the upper-reservoir side, ∂∂x3

u = −p = hU − x3 implies u = 12(hU − x3)2, and similar

condition but with hL instead of hU takes place on the lower-reservoir side.52This follows from (5.117) by using ∂2

∂x23u = − ∂

∂x3p = − ∂

∂x3π+1 = 1 because ∂

∂x3π = v·ν = 0

on the bottom side.


As p ≥ 0 should hold from physical reasons, u should be nonincreasing alongthe x3-direction, hence u ≥ 0. In the dry region one has u = 0, hence 1 −Δu =1 ≥ 0, while in the wet region we derived 1 −Δu = 0 in (5.117). Altogether, weget the following complementarity problem53

−Δu+ 1 ≥ 0, u ≥ 0,

(Δu − 1)u = 0,

}in Ω,

∂∂ν u ≥ 0, u ≥ 0,

u ( ∂∂νu) = 0

}on nonpermeable vertical sides of Γ,

u|Γ prescribed in (5.118a) on the rest of Γ.

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭(5.120)

The corresponding weak formulation admits a unique solution u ∈ K := {v ∈W 1,2(Ω); v|Γ satisfying (5.118a)}, which can be proved straightforwardly by thedirect method as in Theorem 5.3(iii) with Φ(u) :=

∫Ω

12 |∇u|2 − u dx for u ≥ 0

a.e. in Ω, otherwise Φ(u) = +∞. Therefore, the original problem has a unique(very weak) solution p = − ∂

∂x3u ∈ L2(Ω).

5.6.2 Continuous casting: a non-potential variational inequality

A great amount of steel is nowadays casted continuously: hot liquid steel is con-tinuously filled from the top into a mold cooled by water (cf. Figure 14(left)),partly solidifies but keeping still a hot liquid kernel, and continuously extractedby rollers and further cooled down to a complete solidification and then cut toa final product. We shall present only a very simple steady-state model of thisadvanced technology.54 The following notation will be used, cf. also Figure 14below:

θ0 temperature of the liquid phase (melting temperature),θ1 final temperature of the cooled outlet,55

θ2(x) temperature of the environment,

53Note that the Neumann condition (5.118b) is replaced by the complementarity condition onthe nonpermeable vertical sides of Γ, but these are equivalent with each other if u is regularenough because, in the dry region, u = 0 implies ∇u = 0 hence ν · ∇u = ∂

∂νu = 0 on the dry

boundary while on the wet boundary u > 0 and u ( ∂∂ν

u) = 0 imply ∂∂ν

u = 0.54Our simplifications involve, in particular, calm liquid phase on the melting temperature

(i.e. we neglect convection in the liquid part like in Section 6.2), linear heat equation (i.e. weneglect Stefan-Boltzmann radiation on the boundary like (2.125) and temperature dependence ofc and κ), solidification at a single temperature (i.e. no over-cooling effects, no mutual influence ofthe melting temperature and chemical composition of a steel which is, in fact, a mixture of ironand other elements such as carbon, etc.), known temperature θ2 at the mold side (temperaturedistribution in the mold is not solved), etc. Besides a huge amount of papers, the reader isreferred to a monograph by Rodrigues [354, Sect. 2.5].

55This will represent a Dirichlet boundary condition on the bottom end (cf. Figure 14(left))which, however, is rather artificial and simplifies the heat convection in the continuation of thecasted workpiece. Yet, this does not essentially influence the process in the upper part if v3 islarge enough and the bottom end is far enough from the mold.


b ≥ 0 the heat-convection coefficient,

�v = (0, 0, v3) extraction velocity,

κ > 0 the heat-conductivity coefficient,

c > 0 the heat-capacity coefficient,

≥ 0 the latent heat,

x3 = ϕ(x1, x2) a free boundary between the liquid and the solid phases.

Naturally, we assume θ1 < θ0, θ2(x) ≤ θ0, and v3, κ, c and positive. The equationfor the temperature θ in the steady-state extraction regime is:

c�v · ∇θ = κΔθ if θ < θ0. (5.121)

The so-called Stefan condition on the free boundary expresses that the normalheat flux −κ∇θ · ν = κ ∂

∂ν θ is spent as the heat needed for the phase change, herethe solidification, �v · ν:

−κ∂θ

∂ν= − �v · ν, (5.122)

where ν is the unit normal oriented from the liquid phase to the solid one. As alsoθ = θ0 on the free boundary, we have seemingly too many conditions on it but, asin Section 5.6.1, again the position of the free boundary itself is unknown and isto be determined just in this way that both (5.122) and θ = θ0 are fulfilled. As theheat equation is considered only in one phase (here solid) while the temperature ofthe other is assumed constant, this problem is called a one-phase Stefan problem.

The other boundary conditions are outlined in the left-hand part of Fig-ure 14, in particular the conditions on the vertical boundary reflect the cooling byconvection:

−κ∂θ

∂ν= b(x)

(θ − θ2(x)

). (5.123)

FREEBOUNDARY

OF LIQUID STEEL

STEEL

STEEL

WATER

LIQUID

SOLID

MOLD

AIR

CONTINUOUS REFILL

DRIVING

RO

LLERS

extractionvelocity �v = (0, 0, v3)

u=0

u=0

u>0

x1

x3

θ=θ1

θ=θ0

ΩL

ΩSΓSL

κ ∂θ∂ν

+bθ = bθ2(x)

κ∂u∂ν

+bu = bh(x)

κ∂u∂ν

= θ1

Figure 14. Geometric configuration (as a cross-section) of the continuous casting prob-lem and boundary conditions; original (left) and transformed (right).


In terms of the auxiliary function ϕ = ϕ(x1, x2) describing the free boundary inthe sense that ΓSL = {x = (x1, x2, x3) ∈ Ω; x3 = ϕ(x1, x2)}, the condition (5.122)on the free boundary reads as

∂θ

∂x3− ∂θ

∂x1

∂ϕ

∂x1− ∂θ

∂x2

∂ϕ

∂x2=

v3κ

. (5.124)

Formally, we have

κΔθ − c�v · ∇θ = v3∂

∂x3χΩS (5.125)

in the sense of distributions. Indeed, for ΩL:= Ω \ ΩS and ΓSL := ΩS ∩ ΩL (=the

free boundary), and for any v ∈ D(Ω), by using Green’s formula twice and that∇θ = 0 on ΩL, it holds that

〈Δθ, v〉 = −∫Ω

∇θ ·∇v dx = −∫ΩS

∇θ ·∇v dx =

∫ΩS

Δθv dx−∫ΓSL

∂θ

∂νv dS (5.126)

and, again by using Green’s formula twice,⟨ ∂

∂x3χΩS , v

⟩= −

∫Ω

χΩS

∂v

∂x3dx = −

∫ΩS

∂v

∂x3dx =

∫ΓSL

v ν3 dS (5.127)

so that, by using successively (5.126), (5.121), (5.122), and (5.127), one obtains

⟨κΔθ − c�v · ∇θ, v

⟩=

∫ΩS

(κΔθ − c�v · ∇θ)v dx−∫ΓSL

∂θ

∂νv dS

= −∫ΓSL

�vνv dS = − v3

∫ΓSL

v ν3dS = v3

⟨ ∂

∂x3χΩS , v

⟩. (5.128)

Then we use the Baiocchi transformation:

u(x) ≡ u(x1, x2, x3) :=

⎧⎨⎩∫ ϕ(x1,x2)

x3

θ(x1, x2, ξ) dξ for x3 ≤ ϕ(x1, x2),

0 for x3 > ϕ(x1, x2).

(5.129)

Then obviously θ = − ∂∂x3

u and, assuming that θ > 0 in physically relevant situa-tions, Ω

S:= {x∈Ω; u(x) > 0} = {x∈Ω; θ(x) < θ0}. Realizing that �v = (0, 0, v3),

(5.125) transforms by integration in the x3-direction to

κΔu− c�v · ∇u = v3χΩS . (5.130)

Altogether:

κΔu− c�v · ∇u =

{ v30 < v3

and u

{> 0 on ΩS,= 0 on ΩL.

(5.131)


Since the Baiocchi transformation commutes with “ ∂∂ν ” on the vertical lines, the

boundary condition (5.123) transforms to

−∂u

∂ν= bu− h(x), h(x1, x2, x3) =

∫ 0

x3

bθ2(x1, x2, ξ) dξ, (5.132)

provided b is independent of x which we have to assume from now on. The otherboundary conditions are outlined on Figure 14(right). This means we get thecomplementarity problem

−κΔu+ c�v · ∇u ≥ − v3, u ≥ 0,(κΔu− c�v · ∇u+ v3

)u = 0,

⎫⎬⎭ in Ω,

∂u

∂ν+ bu ≥ h, u ≥ 0,(∂u

∂ν+ bu− h

)u = 0

⎫⎪⎬⎪⎭ on Γ.

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭(5.133)

As in Proposition 5.9, we arrive at the variational inequality formulation:∫Ω

κ∇u · ∇(v−u) +(c�v · ∇u+ v3

)(v−u) dx+

∫Γ

(bu− h)(v−u) dS ≥ 0 (5.134)

for v ∈ K := {v ∈W 1,2(Ω); v ≥ 0, v(x1, x2, 0) = 0}. This variational inequalityinvolves a pseudomonotone (non-potential) operator and has a solution u ∈ K byCorollary 5.19; thus we get w = ∂

∂x3u ∈ L2(Ω) a very weak solution. Moreover,

this operator is even uniformly monotone because, by Green’s formula,∫Ω

�v · ∇(u1−u2)(u1−u2) dx =1

2

∫Ω

�v · ∇(u1−u2)2dx

= −1

2

∫Ω

div(�v)(u1−u2)2dx+

1

2

∫Γ

(�v · ν)(u1−u2)2dS ≥ 0; (5.135)

note that the last volume integral vanishes since div(�v) = 0 while the last boundaryintegral is non-negative since (u1−u2)

2 = 0 on top, (�v ·ν) = 0 on vertical sides, andboth (�v ·ν) ≥ 0 and (u1−u2)

2 ≥ 0 on the bottom. Then we can use Theorem 5.20which gives even uniqueness of this solution and continuous dependence on , v3,and h. Example 4.30 showed that this problem is indeed non-potential.


Subdifferentials of convex functions has been scrutinized in many monographsfrom so-called convex analysis, among them Hu and Papageorgiou [209, Sect.3.4],Rockafellar and Wetts [353], or Zeidler [427, Chap.47].

Variational inequalities are addressed in many monographs: Baiocchi, Capelo[30], Chipot [99], Elliott, Ockendon [135], Friedman [155], Glowinski, Lions,


Tremolieres [182], Goeleven, Motreanu [183], Kinderlehrer, Stampacchia [232], Li-ons [261, Chap.2,Sect.8 and Chap.3,Sect.5], Maly, Ziemer [271, Chap.5-6], Pascali,Sburlan [325], Rodrigues [354], Ruzicka [376, Sect. 3.3.4], Troianiello [409], and Zei-dler [427, Chap.54]. A fundamental paper is by Brezis [65, Chap.I]. Applicationsto mechanics, in particular to contact problems, is in Duvaut, Lions [130], Eck,Jarusek, Krbec [132], Hlavacek, Haslinger, Necas, Lovısek [204], Necas, Hlavacek[308], or Kikuchi and Oden [231]. Variational inequalities in the context of theiroptimal control are in Barbu [38, Chap.3] and Outrata, Kocvara, and Zowe [321].

Quasivariational inequalities have been thoroughly exposed in the mono-graph by Baiocchi and Capelo [30], cf. also Aubin [27, Sect. 9.11]. Importantapplication is the ground-water propagation through a dam of a general, non-rectangular shape, see Baiocchi and Capelo [30, Chap.8], Chipot [99, Chap.8], orCrank [111, Sect.2.3.7]. A related subject (not mentioned here) is the so-calledimplicit variational inequalities: find u such that, for all w ∈ V , it holds thatA(u,w)−A(u, u) + F (E(u), w) − F (E(u), u) ≥ 0. Typically, it involves problemslike mechanical contacts with friction that have a dual formulation as quasivari-ational inequalities. Transformation between it and quasivariational inequality isin Mosco [294].

For hemivariational inequalities, introduced essentially by Panagiotopoulos[322] with a certain motivation in continuum mechanics, see also the monographsby Goeleven and Motreanu [183], Haslinger, Mietinen, and Panagiotopoulos [199]and Naniewicz and Panagiotopoulos [298].

A generalization for the monotone set-valued part being non-potential doesexist, too, being based on the concept of the maximal monotone set-valued map-pings. An analog of Browder-Minty’s theorem says that any maximal monotoneand coercive A : V ⇒ V ∗ is surjective, i.e. the inclusion A(u) � f has at leastone solution for any f ∈ V ∗; cf. Hu and Papageorgiou [209, Sect.3.1-2] or Zeidler[427, Chap.32]. Set-valued generalization does exist also for pseudomonotone map-pings56, being invented by Browder [76]. Set-valued generalization of mappings oftype (M)57 is due to Kenmochi [227]. For the surjectivity of pseudomonotone set-valued mappings we refer to Browder and Hess [77]; a thorough exposition is inthe handbook by Hu and Papageorgiou [209, Part I, Chap.III].

56A set-valued mapping A : V ⇒ V ∗ is called pseudomonotone if1) ∀u ∈ V : A(u) is nonempty, bounded, closed, and convex,2) ∀U ⊂ V finite-dimensional subspace: A|U is (norm,weak*)-upper semicontinuous,3) if uk⇀u, fk∈A(uk), lim sup

k→∞〈fk, uk−u〉≤0, then ∀v∈V ∃f∈A(u): lim inf

k→∞〈fk , uk−v〉≥〈f, u−v〉.

57A set-valued mapping A : V ⇒ V ∗ is called of type (M) if A|U is weakly* upper semicon-tinuous for all U ⊂ V finite-dimensional, A(u) is nonempty, bounded, closed, and convex, and iffk ∈ A(uk), and (uk, fk)

∗⇀ (u, f) in V × V ∗ and limsupk→∞〈fk , uk〉 ≤ 〈f, u〉, then f ∈ A(u).

Chapter 6

Systems of equations: particularexamples

No general theory for systems of nonlinear equations exists. Systems usually re-quire a combination of specific, sometimes very sophisticated tricks, possibly witha fixed-point technique finely fitted to a particular structure. Although certaingeneral approaches can be adopted,1 a pragmatic observation is that systems aremuch more difficult than single equations and sometimes only partial results (typ-ically for small data) can be obtained with current knowledge. Even worse, manynatural systems arising from physical problems still remain unsolved with respectto even the existence of a solution; in particular cases, however, this may be relatedwith an oscillatory-like or explosion-like character of related evolutionary systemswhich thus lack any steady states that would solve these stationary systems.

We confine ourselves to only a few illustrative examples having a straight-forward physical interpretation and using the previously exposed theory in a non-trivial but still rather uncomplicated manner.

6.1 Minimization-type variational method: polyconvex

functionals

For the “Lagrangean” ϕ : Ω×Rm×Rm×n → R we consider the system of nonlinearequations (j = 1, . . . ,m):

−n∑

i=1

∂

∂xi

∂ϕ

∂Sij(x, u,∇u) +

∂ϕ

∂Rj(x, u,∇u) = gj on Ω,

n∑i=1

νi∂ϕ

∂Sij(x, u,∇u) + bj(x, u) = hj on Γ,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (6.1)

1Cf. Ladyzhenskaya and Uraltseva [250, Chap.8].

171T. Roubí ek, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics 153, DOI 10.1007/978-3-0348-0513-1_ , © Springer Basel 2013 6

172 Chapter 6. Systems of equations: particular examples

where, instead of the notation (r, s) ∈ R×Rn, we used here (R,S) ∈ Rm×Rm×n,and then ϕ = ϕ(x,R, S), like we already did in Sect. 2.4.4; thus S ∈ Rm×n denoteshere a matrix, hopefully without confusion with S occuring in the surface measuredS. The weak formulation of (6.1) is obtained by multiplying the equation in (6.1)by vj , integrating over Ω, summing it for j = 1, . . . ,m, and using Green’s formula:∫

Ω

(∂ϕ∂S

(x, u,∇u) : ∇v +

m∑j=1

∂ϕ

∂Rj(x, u,∇u)vj

)dx

+

m∑j=1

∫Γ

bj(x, u)vj dS =

m∑j=1

( ∫Ω

gjvj dx+

∫Γ

hjvj dS)

(6.2)

for all v ∈ C1(Ω;Rm), where S : S :=∑n

i=1

∑mj=1 SijSij . Assuming still

∂bi∂Rj

=∂bj∂Ri

, i, j = 1, . . . ,m, (6.3)

the left-hand-side of the boundary-value problem (6.1) has a potential

Φ(u) =

∫Ω

ϕ(x, u,∇u) dx+

∫Γ

ψ(x, u) dS, (6.4)

where ψ(x,R) is defined by the formula (cf. (4.23c)):

ψ(x,R) =

∫ 1

0

R · b(x, tR) dt , R∈Rm, b :Γ×Rm → Rm. (6.5)

Although it is, in general, not possible to pass to a limit through a nonlinearity by aweak convergence, cf. Remark 2.39, it is sometimes possible in special nonlinearities(here the determinant) if special sequences (here generated by gradients2) areconsidered:

Lemma 6.1. Let uk ⇀ u in W 1,p(Ω;Rm), p > n, n = m. Then

det∇uk ⇀ det∇u in Lp/n(Ω). (6.6)

Proof. It is a well-known fact from matrix algebra that, for S ∈ Rm×n with m = n,it holds that (

detS)I =

(cof S)S, (6.7)

where the “cofactor” [cofS]ij is the determinant of the matrix arising from S byomitting the ith row and jth column but multiplied by (−1)i+j . Putting S := ∇uand summing it for i, j = 1, . . . , n, this allows us to show

n det∇u =

n∑i,j=1

∂ui

∂xj(cof∇u)ij =

n∑i,j=1

∂

∂xj

(ui(cof∇u)ij

)− ui ∂

∂xj(cof∇u)ij . (6.8)

2Such sequences are rotation free due to the well-known fact that rot(∇u) ≡ 0. This constraintcauses sometimes surprising effects, e.g. concerning higher integrability, cf. Muller [296].

6.1. Minimization-type variational method: polyconvex functionals 173

The last term vanishes because of Piola’s identity∑n

j=1∂

∂xj(cof∇u)ij = 0 for all

i = 1, . . . , n.3

Then, by using subsequently (6.8), twice Green’s formula, and again (6.8),one gets

limk→∞

∫Ω

(det∇uk)v dx =1

nlimk→∞

∫Ω

n∑i,j=1

∂

∂xj

(uik

(cof∇uk

)ij

)v

= − 1

nlimk→∞

n∑i,j=1

∫Ω

uik

(cof∇uk

)ij

∂v

∂xjdx

= − 1

n

n∑i,j=1

∫Ω

ui(cof∇u)ij∂v

∂xjdx =

∫Ω

(det∇u)v dx (6.9)

for any v ∈ D(Ω) because uk → u in L∞(Ω;Rn) � W 1,p(Ω;Rn) and cof∇uk ⇀cof∇u in Lp/(n−1)(Ω;Rn×n) which is obvious for n = 2 while it follows by inductionif n ≥ 3. �

Lemma 6.2 (Weak lower semicontinuity). Let m = n, let ϕ be coercive inthe sense ϕ(x,R, S) ≥ ε|S|p for p > n and ϕ(x,R, ·) be polyconvex in the sense

ϕ(x,R, S) = f(x,R, S, detS) (6.10)

with some f : Ω×Rm×Rm×n×R → R such that f(x,R, ·, ·) is convex and smooth4,and satisfy the following growth conditions:

∃γ ∈ L1(Ω) :∣∣ϕ(x,R, S)

∣∣ ≤ C(γ(x) + β(|R|) + |S|p), (6.11a)

∃γ ∈ Lp′(Ω) :

∣∣∣ ∂f∂S

∣∣∣ ≤ C(γ(x) + β(|R|) + |S|p−1

), (6.11b)

∃γ∈L(p/n)′(Ω) :∣∣∣ ∂f

∂detS

∣∣∣ ≤ C(γ(x) + β(|R|) + |S|p−n

), (6.11c)

for some C ∈ R+ and β : R → R continuous (with arbitrary growth), and moreoverb(x, ·) : Rn → Rn is monotone for a.a. x ∈ Γ and satisfies the growth condition

∃γ∈L1(Γ); |b(x,R)| ≤ γ(x) + β(|R|). (6.12)

Then Φ is weakly lower semicontinuous.5

3See Ciarlet [93, proof of Thm.1.7-1] or Evans [138, Sect.8.1.4, Lemma 1] for technical details.4Differentiability of f(x,R, ·, ·) is just a technical assumption which can be avoided when

selecting (in a measurable way) subgradients of f(x,R, ·, ·) in place of partial derivatives usedhere.

5Recall again our convention that by semicontinuity, see (1.6), we mean the “sequential”semicontinuity. Here, however, it is even equivalent with general-topological semicontinuity whichuses generalized sequences (nets) since Φ is coercive and V separable.


Proof. Take a sequence uk ⇀ u in W 1,p(Ω). By Banach-Steinhaus Theorem 1.1,{uk}k∈N is bounded. Without loss of generality we can suppose that Φ(uk) →lim infk→∞ Φ(uk). We are to show that Φ(u) ≤ limk→∞ Φ(uk).

By the compact embedding W 1,p(Ω) � L∞(Ω;Rm) (recall that n < p isassumed) we have uk → u uniformly on Ω. Let us put

Ωε :={x∈Ω; |∇u(x)| ≤ 1

ε

}. (6.13)

Then limε→0measn(Ω \Ωε) = 0. Using subsequently non-negativity of ϕ, polycon-vexity of ϕ(x,R, ·) hence convexity of f(x,R, ·, ·), one obtains

lim infk→∞

∫Ω

ϕ(x, uk,∇uk) dx ≥ lim infk→∞

∫Ωε

ϕ(x, uk,∇uk) dx

= lim infk→∞

∫Ωε

f(x, uk,∇uk, det∇uk) dx ≥ limk→∞

∫Ωε

f(x, uk,∇u, det∇u) dx

+ limk→∞

∫Ωε

∂f

∂S(x, uk,∇u, det∇u) : (∇uk −∇u) dx

+ limk→∞

∫Ωε

∂f

∂detS(x, uk,∇u, det∇u)(det∇uk − det∇u) dx

=

∫Ωε

f(x, u,∇u, det∇u) dx =

∫Ωε

ϕ(x, u,∇u) dx;

we used the convergences f(x, uk,∇u, det∇u) → f(x, u,∇u, det∇u) inL1(Ωε),

∂∂S f(x, uk,∇u, det∇u) → ∂

∂S f(x, u,∇u, det∇u) in Lp′(Ωε) and

∂∂detS f(x, uk,∇u, det∇u) → ∂

∂detS f(x, u,∇u, det∇u) in L(p/n)′(Ωε) by continuityof the respective Nemytskiı mappings, and eventually ∇uk−∇u ⇀ 0 and, byLemma 6.1, det∇uk−det∇u ⇀ 0. Finally we pass to the limit with ε → 0by using Lebesgue’s Theorem 1.14 to show that the last integral approaches∫Ωϕ(x, u,∇u) dx.

As to the boundary integral, (6.12) makes u �→ ∫Γ ψ(x, u) dS with ψ from

(6.5) continuous and even smooth, and monotonicity of b(x, ·) makes ψ(x, ·) convex,hence the weak lower-semicontinuity follows as in (4.5).

Altogether, the weak lower-semicontinuity of Φ was thus proved. �Proposition 6.3 (Existence: the direct method). Let the assumptions ofLemma 6.2 be valid and, moreover, b be coercive in the sense b(x,R)·R ≥ ε|R|q−kwith ε > 0 and q > 1. Then (6.1) has a weak solution.

Proof. Analogous to Proposition 4.16 with f defined by 〈f, v〉 := ∫Ωg ·v dx+∫

Γh ·

v dS, but simplified due to absence of Dirichlet boundary conditions here. �Remark 6.4 (Polyconvexity). The formula (6.10) gives a good generality only ifm = n = 2. In general, one should assume

ϕ(x,R, S) = f(x,R, (adjiS)

min(n,m)i=1

)(6.14)


with some f : Ω×Rm×∏min(n,m)i=1 Rk(i,n,m) → R, where k(i, n,m) is the number of

all minors of the i-th order, such that f(x,R, ·) is convex, where adjiS denotes thedeterminants of all (i×i)-submatrices. Then, following Ball [31], ϕ(x,R, ·) is calledpolyconvex. In particular, adj1S = S and adjmin(n,m)−1S = cofS and, if m = n,adjmin(n,m)S = detS. Then Lemma 6.1 is to be generalized for adji∇uk ⇀ adji∇u

in Lp/i(Ω;Rk(i,n,m)) provided p > i ≤ min(m,n), and Lemma 6.2 as well asProposition 6.3 is to be modified for (6.14) in place of (6.10).

Remark 6.5 (Quasiconvexity). Polyconvexity of ϕ(x,R, ·) is only sufficient for theweak lower semicontinuity of Φ but not necessary if min(n,m) ≥ 2. The pre-cise condition (i.e. sufficient and necessary) is the so-called W 1,p-quasiconvexity,defined in a rather non-explicit way by

ϕ(x,R, S) = infv∈W 1,p

0 (O;Rm)

1

|O|∫O

ϕ(x,R, S+∇v(ξ)

)dξ (6.15)

where O ⊂ Rn is a (in fact, arbitrary) Lipschitz domain. This condition, whoseinevitable nonlocality has been proved by Kristensen [242], cannot be verifiedefficiently except for very special cases, as e.g. polyconvexity which is a (strictly)stronger condition. Henceforth, another mode, a so-called rank-one convexity, wasintroduced by Morrey [291] by requiring t �→ ϕ(x,R, S + ta ⊗ b) : R → R to beconvex for any a ∈ Rn, b ∈ Rm, [a ⊗ b]ij := aibj . Since Morrey [290] inventedquasiconvexity, the question of coincidence with rank-one convexity was open formany decades and eventually answered negatively by Sverak [401] at least if m ≥ 3and n ≥ 2. For smooth ϕ(x,R, ·), the rank-1 convexity is equivalent with the so-

called Legendre-Hadamard condition ϕ′′S(x,R, S)(S, S) ≥ 0 for all S, S ∈ Rm×n

with S = a⊗b, a ∈ Rm, b ∈ Rn. Obviously, polyconvexity (and thus all mentionednotions) is weaker than usual convexity, and for min(n,m) = 1 all mentionedmodes coincide with usual convexity of ϕ(x,R, ·).Remark 6.6 (Symmetry conditions6). Considering the general system of m quasi-linear equations

−div(aj(x, u,∇u)

)+ cj(x, u,∇u) = gj, j = 1, . . . ,m, (6.16)

the symmetry condition (4.21) which, here together with (6.3), ensures existenceof a potential in the form (6.4) bears now the form

∂ali(x,R, S)

∂Sjk=

∂ajk(x,R, S)

∂Sli, (6.17a)

∂ali(x,R, S)

∂Rj=

∂cj(x,R, S)

∂Sli, (6.17b)

∂cj(x,R, S)

∂Rl=

∂cl(x,R, S)

∂Rj(6.17c)

6See e.g. Necas [305, Sect.3.2].


for all i, k = 1, . . . , n and j, l = 1, . . . ,m and for a.a. (x,R, S) ∈ Ω× Rm × Rm×n.Then, as in (4.23b), ϕ occurring in (6.4) is given (up to a constant) by the formula

ϕ(x,R, S) =

∫ 1

0

m∑j=1

( n∑i=1

aji (x, tR, tS)Sji + cj(x, tR, tS)Rj

)dt (6.18)

and (6.16) coincides with the equation in (6.1) because ∂ϕ/∂Sij = aji and∂ϕ/∂Rj = cj . Like (4.21)–(4.22), now (6.17) expresses just symmetry of the Jaco-bian of the mapping (R,S) �→ (c(x,R, S), a(x,R, S)) : Rm×Rm×n → Rm×Rm×n.In this case, (6.16) is the Euler-Lagrange equation for the potential having the“density” (6.18).

Example 6.7 (Elasticity: large strains). Systems (6.1) with ϕ(x,R, S) = φ(x, I+S)occur in steady-state elasticity where n = m, u : Ω → Rn means displace-ment of a body occupying in an undeformed state the reference domain Ω whiley(x) := x + u(x) defines the deformation at x∈Ω. The deformed body then oc-cupies the domain y(Ω) ⊂ Rn and φ(x, F ) expresses the specific stored energy atx ∈ Ω and at the deformation gradient F = I + S. The direct method used inProposition 6.3 expresses minimization of overall stored energy and energy con-tained in an elastic support on the boundary (through ψ) which is a variationalprinciple that sometimes (but not always) governs steady states of loaded elasticbodies. The so-called frame-indifference principle requires φ(x, ·) in fact to dependonly on the so-called (right) Cauchy-Green stretch tensor

C = FF = (S + I)(S + I) = I+ S + S + SS. (6.19)

The often considered potential

φ(x, F ) :=1

2ECE, E :=

1

2(C − I), C = FF, (6.20)

where E is called the Green-Lagrange strain tensor, describes the so-calledSaint Venant-Kirchhoff’s material with C = [Cijkl ] the positive-definite elastic-moduli tensor. This 4th-order tensor C has a lot of symmetries leading to only fewindependent entries.7 Unfortunately, the choice (6.20) leads to ϕ(x,R, ·) which iseven not rank-one convex, however. An example of a polyconvex energy ϕ(x,R, ·)is Mooney-Rivlin’s material described by8

φ(x, F ) := c1tr(E) + c2tr(cof(C)−I

)+ φ0

(det(F )

), (6.21)

with C,E again from (6.20), c1, c2 > 0 and φ0 a convex function; tr(·) in (6.21)denotes the trace of a matrix. This is a special case of a so-called Ogden’s material.9

7Number of independent entries of C in case of anisotropic crystals: 3 (cubic), 6 (tetragonal),9 (orthorombic), 13 (monoclinic), or 21 (triclinic). Polycrystalic materials can be consideredisotropic and leads to 2 independent entries only, cf. (6.23).

8Note that det(F ) = det(F) =√

det(FF ) =√

det(C) actually depends only on C hence(6.21) leads indeed to a frame-indifferent potential.

9Ogden’s material allows for more general nonlinearities, cf. e.g. Zeidler [427, Sect.61.8 and62.14]. In this way, the coercivity in Lemma 6.2 can be satisfied.


Example 6.8 (Elasticity: small strains). If the displacement u is small, one canneglect the higher-order term SS in (6.19) so that the Green-Lagrange straintensor E from (6.20) turns into a so-called small-strain tensor e(u) := 1

2∇u +12 (∇u), i.e.

eij(u) =1

2

∂ui

∂xj+

1

2

∂uj

∂xi. (6.22)

In fact, only the gradient of u is to be small rather than u itself. Then theSt.Venant-Kirchhoff’s potential φ from (6.20) with e(u) substituted for E turns ϕinto a quadratic form of the displacement gradient ∇u. For isotropic material, itlooks as

ϕ(x,R, S) = ϕ(S) = μ∣∣∣12S +

1

2S∣∣∣2 + λ

2

(tr(S)

)2, (6.23)

i.e. ϕ(∇u) = μ|e(u)|2 + 12λ(div u)

2, where μ > 0 and λ ≥ 0 stand here for the so-called Lame constants describing the elastic response on shear and compression,respectively. In particular, ϕ is then convex and (6.1) reduces to a so-called Lamesystem of linear elasticity whose weak formulation (6.2) then results in10∫

Ω

σ(e(u)):e(v) dx +

∫Γ

b(u) · v dS =

∫Ω

g · v dx+

∫Γ

h · v dS (6.24)

where σ(e(u)) denotes the stress tensor[σ(e(u))

]ij= 2μeij(u) + λ(div u)δij (6.25)

with δij the Kronecker symbol.

Remark 6.9 (Bibliographical notes). In fact, polyconvexity provides existenceproof on W 1,p(Ω;R3) even for p ≥ 3 and faster growth of ϕ admitting ϕ → +∞if det(I +∇u) ↘ 0, see Ball [31]. For advanced study of deep and difficult topicsaround quasiconvexity the reader is referred to monographs by Dacorogna [112,Chap.IV], Evans [138, Chap.8], Giaquinta, Modica and J. Soucek [177, Part II,Sect.1.4], Giusti [180, Chap.5], Morrey [291], Muller [297], and Pedregal [331,Chap.3]. Existence of a minimizer of (6.4) on W 1,p(Ω;Rm) is due to Acerbi andFusco [2]. For mathematical aspects of nonlinear elasticity see monographs byCiarlet [93], Pedregal [332], or Zeidler [427, Vol.4]. Although elasticity theory hasreceived attention throughout centuries, there are still many open fundamentalproblems in nonlinear elasticity especially when ϕ(x,R, ·) has faster growth thanpolynomial11, see [32]. E.g., a question about injectivity of y : Ω → Rn, in partic-ular avoiding self-contact, has been pointed out by Ciarlet and Necas [94]. Linearelasticity is exposed e.g. in Duvaut and Lions [130] or Necas and Hlavacek [308],and related unilateral problems in Hlavacek et al. [204].

10Note that σ(e(u)):∇v = σ(e(u)):e(v) + 12σ(e(u)) : (∇v − (∇v)) and the last term vanishes

because σ(e(u)) is symmetric and thus orthogonal to antisymmetric matrices.11It is quite natural to assume especially ϕ(x,R, S)→ +∞ when det(I+ S)↘ 0.


6.2 Buoyancy-driven viscous flow

It is an every-day experience that a warmer fluid in the gravity field tends torun up while a cooler fluid falls down, in special situations known as Benard’sproblem12. These processes obviously involve mutually coupled velocity and tem-perature fields. Oberbeck-Boussinesq’s model for (a steady-state of) this processinvolving incompressible viscous non-Newtonean fluid13 occupying a fixed domainΩ is governed by the following system14:

(u · ∇)u − div σ(e(u)

)+∇π = g(1− αθ), (6.26a)

div u = 0 , (6.26b)

u · ∇θ − κΔθ = 0 , (6.26c)

with e(u) := 12 (∇u) + 1

2∇u as in (6.22) and where we denotedu : Ω → Rn a velocity field,π : Ω → R a pressure field,θ : Ω → R a temperature field,κ > 0 the heat-conductivity coefficient,α a coefficient of mass density variation with respect to temperature,σ =σ(e) = the viscous stress tensor,g = an external (e.g. gravity) force.

We have to specify boundary conditions. Let us consider, e.g., no-slip for u andNewton’s condition for θ with some b1 > 0:

u = 0 , κ∂θ

∂ν+ b1θ = h on Γ. (6.27)

Let us assume that, for some 0 < c1 ≤ c3, 0 ≥ c2, and q := 2∗p∗/(2∗p∗−p∗−2∗):

∀e ∈ Rn×nsym : σ(e):e ≥ c1|e|p, |σ(e)| ≤ c2|e|p−1+c3, (6.28a)

∀e1, e2 ∈ Rn×nsym , e1 �= e2 :

(σ(e1)−σ(e2)

): (e1−e2) > 0, (6.28b)

g ∈ Lq(Ω;Rn), h ∈ L2#′(Γ), (6.28c)

where Rn×nsym denotes the set of n×n symmetric matrices. An example for (6.28a)–

(6.28b) is σ(e) = |e|p−2e.We will employ a fixed-point technique, which illustrates quite a typical ap-

proach to systems of equations and works here easily also for p ≤ 2 in contrast totechniques based on joint coercivity of this system, cf. Exercise 6.19 below. Let usdenote

W 1,p0,div(Ω;R

n) := {v ∈ W 1,p0 (Ω;Rn); div v = 0} (6.29)

12Cf. Straughan [392, Chap.3].13The adjective “non-Newtonean” refers to a non-constant viscosity; cf. Remark 6.15.14This model is derived from a full compressible system on assumptions of small u, nearly

constant θ, and negligible dissipative and adiabatic heat. Non-Newtonean fluids in this contexthave been used in Malek at al. [270]. See [392] for an extensive reference list. For a more generalmodel see e.g. [223, 363].

6.2. Buoyancy-driven viscous flow 179

and consider a mapping M from W 1,p0,div(Ω;R

n) to itself, defined by

M := M2 ◦(M1×id

): v �→ u, M1 : v �→ θ, M2 : (v, θ) �→ u, (6.30)

with u and θ being the weak solutions to

(v · ∇)u− div σ(e(u)

)+∇π= g(1− αθ), div u = 0, u|Γ = 0 , (6.31a)

v · ∇θ − κΔθ= 0 , κ∂θ

∂ν+ b1θ|Γ = h . (6.31b)

Note that the system (6.31) is decoupled: first, one can solve (6.31b) to get θ andthen, knowing both v and θ, one can solve (6.31a). If σ is linear, then the problem(6.31a) arising via the “frozen” velocity v in the convective term is also linear andis then called the Oseen equation.

Lemma 6.10 (A-priori estimates). Let p > 1, p ≥ max(n/2, 3n/(n+2)). Thereis R dependent on c1, c2, c3, g and h but not on v ∈ W 1,p

0,div(Ω;Rn) such that

‖θ‖W 1,2(Ω) ≤ R and ‖u‖W 1,p(Ω;Rn) ≤ R. (6.32)

Proof. To estimate the temperature, we test (6.31b) by θ and, for p ≥ n/2,15 useGreen’s Theorem 1.31 and the identity∫

Ω

(v · ∇θ) θ dx =1

2

∫Ω

v · ∇θ2 dx = −1

2

∫Ω

(div v) θ2 dx = 0 (6.33)

so that, by the Poincare inequality (1.56) considered with p = 2 = q,

C−1P

min(κ, b1)‖θ‖2W 1,2(Ω) ≤∫Ω

(v · ∇θ) θ dx+ κ

∫Ω

|∇θ|2 dx+ b1

∫Γ

θ2 dS

=

∫Γ

hθ dS ≤ ‖h‖L2#

′(Γ)

‖θ‖L2#(Γ) ≤N2

4ε‖h‖2

L2#′(Γ)

+ ε‖θ‖2W 1,2(Ω) , (6.34)

where CP comes from (1.56) and N is the norm of the trace operator W 1,2(Ω) →L2#(Γ). For ε > 0 small enough, it gives ‖θ‖W 1,2(Ω) ≤ R with R independent of v.

To estimate the velocity, we test (6.31a) by u and use, for p ≥ 3n/(n+2),Green’s Theorem 1.31 and the identities

∫Ω ∇π · u dx = − ∫Ω πdiv u dx = 0 and16∫

Ω

((v · ∇)u

) · u dx =

∫Ω

n∑k=1

n∑j=1

vk∂uj

∂xkuj dx

= −∫Ω

( n∑k=1

n∑j=1

uj∂vk∂xk

uj + ujvk∂uj

∂xk

)dx = −

∫Ω

((v · ∇)u

) · u dx (6.35)

15To ensure integrability of (v · ∇θ) θ in (6.33) when θ ∈ L2∗ (Ω) and ∇θ ∈ L2(Ω;Rn), oneneeds v ∈ Ln(Ω;Rn), i.e. one needs p∗ ≥ n, which is just equivalent to p ≥ n/2.

16Note that∑n

k=1 ∂vk/∂xk = div v = 0.


so that ∫Ω

((v · ∇)u

) · u dx = 0 (6.36)

and, by Korn’s inequality (1.59) and by (6.28a),

c1C−pK

‖u‖pW 1,p

0 (Ω;Rn)≤ c1‖e(u)‖pLp(Ω;Rn×n) ≤

∫Ω

σ(e(u)) :e(u) dx

=

∫Ω

((v ·∇)u) · u+ σ(e(u)) :e(u) +∇π ·u dx =

∫Ω

g(1−αθ) · u dx

≤ ‖g‖Lq(Ω;Rn)

(measn(Ω)

1/2∗ + α‖θ‖L2� (Ω)

)‖u‖Lp∗(Ω;Rn). (6.37)

From this, the second estimate in (6.32) follows by Young’s inequality and thealready obtained estimate of θ. �Lemma 6.11 (Uniqueness and continuity17). Let p > 3n/(n+2). Given v, thesolution (u, θ) to (6.31) is unique. Besides, M : v �→ u is weakly continuous.

Proof. Uniqueness of temperature θ follows from the a-priori estimate (6.34)because (6.31b) is linear in terms of θ. The weak continuity of M1 is obvi-ous when one realizes that vk·∇θk ⇀ v·∇θ weakly in L1(Ω) because vk ⇀ vweakly in W 1,p(Ω;Rn) (hence strongly in Lp∗−ε(Ω;Rn)) and ∇θk ⇀ ∇θ weaklyin L2(Ω;Rn×n).

For the uniqueness of the velocity, we take u1, u2 two weak solutions of(6.31a), and test the difference of the weak formulation of (6.31a) by u12 := u1−u2.Using (6.36) for u12 instead of u, it gives:∫

Ω

(σ(e(u1))− σ(e(u2))

):e(u12) dx = −

∫Ω

((v · ∇)u12) · u12 dx = 0

so that, by strict monotonicity (6.28b), it holds that e(u12) = 0 a.e. in Ω and then,by Korn’s inequality (1.59), u12 = 0.

To show the weak continuity of M2 : (v, θ) �→ u, one can use monotonicity ofσ and consider a weakly converging sequence (vk, θk) ⇀ (v, θ) and correspondingsolutions uk:

0 ≤∫Ω

(σ(e(uk))− σ(e(w))

):e(uk−w) dx

=

∫Ω

(g(1−αθk)− (vk · ∇)uk

)(uk−w)− σ(e(w)):e(uk−w) dx

→∫Ω

(g(1− αθ) − (v · ∇)u

)(u−w)− σ(e(w)):e(u−w) dx (6.38)

and then use Minty’s trick. Note that (6.38) used the compact embeddingW 1,p

0 (Ω;Rn) � L(p∗−ε)(Ω;Rn) which allowed for the limit passage in the term∫Ω(v

k · ∇uk)ukdx if p−1 + 2(p∗ − ε)−1 ≤ 1 which requires p > 3n/(n+2). �17A more involved technique allows for improving existence for non-Newtonean fluids (6.26a,b)

itself even for p > 2n/(n+2), cf. [122, 151].


Proposition 6.12 (Existence). Let (6.28) hold. Then the system (6.26) has atleast one weak solution.

Proof. It follows from Schauder’s fixed-point Theorem 1.9 (cf. Exercise 2.55) forM on the ball in B := {v ∈ W 1,p

0,div(Ω;Rn); ‖v‖W 1,p(Ω;Rn) ≤ R} with a sufficiently

large radius R from (6.32) depending only on the data g, h, σ, and α, endowed bythe weak topology which makes it compact. �

An alternative to the no-slip (i.e. u = 0) boundary conditions (6.27) is apartial-slip condition (with 0 ≤ γ1 ≤ γ phenomenological coefficients) which, inour thermally coupled system, reads as:

un = 0, �σt+γut = 0, (6.39a)

κ∂θ

∂ν+ b1θ = h+ γ1

∣∣ut

∣∣2 , (6.39b)

where ut := u−un is the tangential velocity and un := (u·ν) ν is the nor-mal velocity, and similarly for the so-called traction force �σ defined as [�σ]i =∑n

j=1 σij(e(u))νj . The two conditions in (6.39a) form the so-called Navier bound-

ary condition.18 For γ = 0, it expresses a no-stick (or ideally slippery) bound-ary while for γ → +∞ it approximates the no-slip boundary. Instead ofW 1,p

0,div(Ω;Rn) defined in (6.29), the weak formulation now uses the linear space

{v∈W 1,p(Ω;Rn); div v = 0 in Ω, ν·v = 0 on Γ} and leads to a weak formulationof (6.26a,b) as the integral identity∫

Ω

((u·∇)u)·z + σ(e(u)):e(z) dx+

∫Γ

γut ·zt dS =

∫Ω

g(1−αθ) dx (6.40)

In contrast to the no-slip or the no-stick cases, for 0 < γ < +∞ the condition(6.39a) dissipates energy which may partly (namely with the ratio γ1/γ ∈ [0, 1])contribute to a heat production on the boundary. This just gives rise to the lastterm in (6.39b). If γ1 < γ, the resting portion 1−γ1/γ of the mechanical dissipatedenergy is indeed lost from the system in this model.

For γ1 > 0, we have an additional coupling, which might be not entirelyeasy to treat and which may even cause doubt about existence of solutions if thefluid is not dissipative enough (i.e. here if p ≤ 3, which is related to the quadraticgrowth of the boundary term γ1|ut|2) and simultaneously if the external supply ofenergy is not small. Let us confine ourselves to the gravity-type force g such thatcurl g = 0, where the rotation is defined as curl · := ∇×· where × : R3×R3 → R3

is the vector product on R3.

Proposition 6.13 (Existence with Navier’s boundary conditions). Let(6.28) hold with p ≥ 4n/(n+4) and let one of the following two conditions holds

18The Navier’s conditions (6.39a) are “mathematically” very natural in comparison with mereDirichlet condition u = 0, as pointed out by Frehse and Malek [150].


(i) p > 3 (i.e. the fluid is “dissipative enough”), or

(ii) 1 < p < 3 and the heat flux h be “small enough” in L2#′(Γ).

Then the system (6.26) with the boundary conditions (6.39) has weak solutions.

Proof. Again, use the Schauder fixed-point theorem based, instead of (6.31) ontwo decoupled problems:

(v·∇)u− div σ(e(u)

)+∇π = g(1−αθ), div u = 0 on Ω,

un = 0 and[σ(e(u))ν

]t+γut = 0 on Γ,

}(6.41a)

v·∇θ − κΔθ = 0 on Ω,

κ ∂θ∂ν + b1θ|Γ = h+ γ1

∣∣vt∣∣2 on Γ.

}(6.41b)

Note that, for v ∈ W 1,p(Ω;Rn), the additional boundary heat source γ1|vt|2 be-

longs to Lp#/2(Γ) ⊂ L2#′(Γ) provided p#/2 ≥ 2#

′, which is fulfilled only if

p ≥ 4n/(n+4), so that (6.41b) possesses a conventional weak solution θ ∈ W 1,2(Ω)which is unique and depends continuously on v. Also, curl g = 0 means exis-tence of a potential ϕ so that g = ∇ϕ, and then

∫Ω g·u dx =

∫Ω∇ϕ·u dx =∫

Γϕu·ν dS − ∫

Ωϕdiv u dx = 0. From (6.41a) tested by u, one gets

‖u‖p−1W 1,p(Ω;Rn) ≤ C1‖θ‖W 1,2(Ω) (6.42)

with C1 depending on c1 from (6.28a) and on g and α. The estimate (6.34) is nowmodified by using

∫Γ(h+γ1|vt|2)θ dS ≤ 2(‖h‖

L2#′(Γ)

+γ1‖v‖2L2#

′2(Γ)

)‖θ‖L2#(Γ) so

that it gives

‖θ‖W 1,2(Ω) ≤ C0

(‖h‖2L2#

′(Γ)

+ ‖v‖2W 1,p(Ω;Rn)

)(6.43)

for some C0 depending also on the norms of the trace operators W 1,p(Ω) →L2#

′2(Γ) and W 1,2(Ω) → L2#(Γ). For p > 3, we can combine (6.42) and (6.43) to

obtain

‖u‖p−1W 1,p(Ω;Rn) ≤ C0C1

(‖h‖2L2#

′(Γ)

+ ‖v‖2W 1,p(Ω;Rn)

) ≤ Cp+1

2‖v‖p−1

W 1,p(Ω;Rn) (6.44)

with some Cp depending on p, C0C1, and h. This suggests that we choose v fromthe ball of the radius (2Cp)

1/(p−1) to obtain the property that M from (6.30) mapsit into itself. Thus (i) has been proved.

As to (ii), we can exploit the first inequality in (6.44) to see that, for suf-ficiently small ‖h‖

L2#′(Γ)

, we can again find r such that ‖v‖W 1,p(Ω;Rn) ≤ r will

imply ‖u‖W 1,p(Ω;Rn) ≤ r. �


Remark 6.14 (Coupling through dissipative/adiabatic heat effects). The Oberbeck-Boussinesq model (6.26) is only one of various options and there are various modi-fications appearing in literature.19 Typically, some enhancement of the heat equa-tion (6.26c) as

u·∇θ − κΔθ = α1σ(e(u)):e(u)− α2θu (6.45)

is considered with some phenomenological coefficients α1 and α2; usually 0 ≤α1 ≤ 1 and 0 ≤ α2 ≤ α is considered. This makes another nonlinear couplingwhich violates coercivity of the whole system, so that one can expect existenceof a solution possibly for small data only. Indeed, assuming p > n, n = 2 or 3, a

smooth domain Ω, and b1(x) ≥ b0 > 0 with b1 ∈ W 1,2#2/(2#−2)(Γ) and c3 = 0 in(6.28a), one can use the fixed point of the mapping (v, ϑ) �→ (u, θ) determined by

(v·∇)u− div σ(e(u)

)+∇π = g(1−αϑ), div u = 0 on Ω,

u = 0 on Γ,

}(6.46a)

v·∇θ − κΔθ = α1σ(e(v)):e(v)− α2ϑv on Ω,

κ ∂θ∂ν + b1θ|Γ = h on Γ.

}(6.46b)

We need to use L1-theory for the heat equation from Chapter 3. Namely, we usethe interpolation for (3.47) with A := κI, and g := α1σ(e(v)):e(v)−α2ϑv combinedwith a bootstrap argument of the advective term v·∇θ with v ∈ W 1,p

0,div(Ω;Rn) to

obtain the estimate20∥∥θ∥∥Lq(Ω)

≤ K(1+‖v‖rW 1,p(Ω;Rn)

)(∥∥α1σ(e(v)):e(v)−α2ϑv∥∥L1(Ω)

+∥∥h∥∥

L1(Γ)

)(6.47)

with q < n/(n−2) and r > n/2−1 for a distributional solution θ to theboundary-value problem (6.46b). Combining (6.47) with the obvious estimate‖u‖p−1

W 1,p(Ω;Rn) ≤ C‖θ∥∥Lq(Ω)

and assuming (6.28a) with c3 = 0, one obtains an

estimate of the type

‖u‖p−1W 1,p(Ω;Rn) ≤ α1C‖v‖pW 1,p(Ω;Rn) + α1C‖v‖p+r

W 1,p(Ω;Rn)

+ α2‖v‖W 1,p(Ω;Rn)

(1+‖v‖rW 1,p(Ω;Rn)

)‖ϑ‖L1(Γ) + ‖v‖rW 1,p(Ω;Rn)‖h‖Lq(Ω). (6.48)

19Cf. e.g. [223, 343] for a genesis of various possibilities. The starting point is always thecomplete compressible fluid system of n+2 conservation laws for mass, impulse, and energy.Then, the so-called incompressible limit represents a small perturbation around a stationaryhomogeneous state, i.e. around constant mass density, constant temperature, and zero velocity.

20The mapping B used in the proof of Lemma 3.29 allows for the estimates‖B‖L (W1,2(Ω)∗,W1,2(Ω)) ≤ C and ‖B‖L (L2(Ω)∗,W2,2(Ω)) ≤ C(1+‖v‖L∞(Ω;Rn)) when realizing

that v∈L∞(Ω;Rn) and v·∇θ∈L2(Ω). Thus, by interpolation, also ‖B‖L (Wλ,2(Ω)∗,W2−λ,2(Ω))

≤ C(1+‖v‖L∞(Ω;Rn))1−λ, which yields (6.47) by transposition used for λ < 2−n/2 as in the

proof of Lemma 3.29, together with the embedding Wλ,2(Ω) ⊂ L2n/(n−2λ)(Ω).


Then, if the external heating h is small in L1(Γ)-norm, one can find a sufficientlysmall ball in W 1,p

0,div(Ω;Rn)×Lq(Γ) mapped to itself by the mapping (v, ϑ) �→ (u, θ)

with u being the weak solution to (6.46a) and θ being the distributional solutionto (6.46b). For such sort of existence analysis see also [299, 363] even for p ≤ n.

Remark 6.15 (Navier-Stokes equations). For σ(e) = 2μe, (6.26a,b) turns (whenneglecting buoyancy, i.e. α = 0, and thus also temperature variation) into thesystem21

(u · ∇)u − μΔu+∇π = g, div u = 0, (6.49)

which is called a steady-state Navier-Stokes system.22 The coefficient μ is calleda kinematic viscosity coefficient and fluids exhibiting such constant viscosity arecalled Newtonean fluids. The no-stick boundary conditions (6.39a) have now analternative (though not fully equivalent) option: u·ν = 0, (curlu)·ν = 0, and(curl2 u)·ν = 0 on Γ, see [42]. For other conditions we refer also to [105].

Exercise 6.16. Write a weak formulation for (6.31), use divergence-free test func-tions.23 Similarly, derive the weak formulation (6.40) of the Navier boundary valueproblem (6.26a,b)–(6.39a).24

Exercise 6.17. Prove Proposition 6.12 when assuming only mere monotonicity ofσ instead of the strict monotonicity (6.28b).25

Exercise 6.18 (Temperature-dependent viscosity26). Modify the proofs of Lem-mas 6.10 and 6.11 if σ = σ(θ, e) in (6.26a) and (6.31a), assuming continuity of σand the properties (6.28a,b) holding for σ(θ, ·) uniformly for θ.

21Indeed, taking into account divu =∑n

i=1 ∂ui/∂xi = 0 and σ(e(u)) = 2μe(u), one gets

div 2μe(u) = 2μn∑

i=1

∂

∂xi

(1

2

∂ui

∂xj+

1

2

∂uj

∂xi

)= μ

(∂

∂xj

( n∑i=1

∂ui

∂xi

)+

n∑i=1

∂2uj

∂xi∂xi

)= μΔu.

22Cf. Constantin and Foias [106], Galdi [170], Sohr [391], or Temam [403] for a thoroughtreatment.

23Hint: Realize that, by Green’s Theorem 1.31,∫Ω∇π · z dx =

∫Ω−π div v dx = 0 and that

symmetric and antisymmetric matrices are mutually orthogonal so that σ symmetric implies∫Ω−div(σ(e(u)))·z dx =

∫Ωσ(e(u)):∇z dx =

∫Ωσ(e(u)):e(z) dx.

cf. also (6.40). As to (6.31b), the advective term leads either to∫Ω v·∇θz dx or to

∫Ω−θv·∇z dx.

24Hint: After applying Green’s formula, treat the resulting boundary term as∫Γσ(e(u)):(z⊗ν) dS =

∫Γ�σ·z dS =

∫Γ

(�σn+�σt

)·(zn+zt)dS =

∫Γ�σn·zn+�σt·zt dS = −

∫Γγut·zt dS

when realizing that normal and tangential vectors are mutually orthogonal, and when usingzn = 0 and the latter condition in (6.39a).

25Hint: Prove that the mapping M2 defined by (6.30), which is now set-valued, has convexvalues (cf. Theorem 2.14(i)) and is upper semi-continuous, and then use Kakutani’s fixed-pointTheorem 1.11 instead of Schauder’s.

26For a non-Newtonean model with viscosity dependent also on temperature we refer to e.g.[270]. The Newtonean case was treated in [300] even for the coupled system as in Remark 6.14.


Exercise 6.19. Considering the boundary-value problem (6.26)–(6.27) and p > 2,verify the assumptions of Brezis theorem 2.6 for V = W 1,p

0,div(Ω;Rn)×W 1,2(Ω).27

Prove existence of u and of θ solving (6.31) by the Galerkin method. Consideringσ(e) = |e|p−1e, show strong convergence of this Galerkin approximation.28 Dothe same for the problem (6.26)–(6.39) for p > 3 by considering V = {(v, θ) ∈W 1,p(Ω;Rn)×W 1,2(Ω); div v = 0 in Ω, ν·v = 0 on Γ}.29Exercise 6.20. Uniqueness of the obtained solution does not hold in generalby natural reasons. Nevertheless, prove uniqueness of the weak solution to(6.26)–(6.27) for g and h small enough provided σ is strongly monotone, i.e.(σ(e1)−σ(e2)):(e1−e2) ≥ c3|e1−e2|2 for some c3 > 0, and provided also n ≤ 4and q ≥ n/2 for (6.28c).30

Exercise 6.21 (Nonlinear heat transfer). Instead of (6.26c), consider the nonlinearheat transfer c(θ)u·∇θ−div(κ(θ)∇θ) = 0 with c and κ denoting the heat capacityand heat-transfer coefficient depending possibly on temperature, cf. also (2.136).Then apply the enthalpy transformation to replace it by u·∇w − Δβ(w) = 0,cf. Remark 3.25, and, denoting still by γ the inverse of the primitive function toc, write the resulting system as

(u·∇)u− div σ(e(u)

)+∇π = g(1− αγ(w)), (6.50a)

div u = 0 , (6.50b)

u·∇w −Δβ(w) = 0 . (6.50c)

27Hint: Show the coercivity of the underlying operator on V by testing (6.31a) and (6.31b)respectively by u and θ, imitating and merging the estimates (6.34) and (6.38). Realize that,if the assumption p > 2 were not hold, there would be difficulties with estimating the term∫Ωgαθu dx.

28Hint: Employ d-monotonicity of the highest-order parts of (6.31) together with uniformconvexity of the underlying space V .

29Hint: Imitate and merge the estimates (6.42) and (6.43).30Hint: Consider two solutions (u1, θ1) and (u2, θ2), test the difference of weak formulations

of (6.26a,b) by u12 := u1 − u2 and of (6.26c) by θ12 := θ1 − θ2, sum them up, realize that smalldata imply both ‖u2‖W1,p(Ω;Rn) and ‖θ2‖W1,2(Ω) small, and estimate∫

Ωκ|θ12|2 + c3|e(u12)|2 dx+

∫Γb1|θ12|2 dS

≤∫Ωgαθ12·u12 − (

(u1·∇)u1−(u2·∇)u2)u12 − (

u1·∇θ1−u2·∇θ2)θ12 dx

=

∫Ωgαθ12·u12 − (

(u1·∇)u12+(u12·∇)u2)u12 − (

u1·∇θ12+u12·∇θ2)θ12 dx

=

∫Ωgαθ12·u12 − (

(u12·∇)u2)u12 − (

u12·∇θ2)θ12 dx

≤ α‖g‖Lq(Ω;Rn)‖θ12‖L2∗ (Ω)‖u12‖L2∗ (Ω;Rn) + C1‖u12‖2L2∗ (Ω;Rn)

‖∇u2‖Lp(Ω;Rn×n)

+C2‖∇θ2‖L2(Ω;Rn)‖u12‖L2∗ (Ω;Rn)‖θ12‖L2∗ (Ω)

with C1, C2 depending on p and n, and finally by Young, Poincare and Korn inequalities concludethat θ12 = 0 and u12 = 0. The condition n ≤ 4 is needed for Holder’s inequality leading to C2.


Considering again the no-slip boundary conditions (6.27), now in the form u = 0and ∂

∂νβ(w)+ b1γ(w) = h, design suitably a mapping whose fixed point does existand is a weak solution to the system (6.50).31

6.3 Reaction-diffusion system

Let us investigate the so-called steady-state Lotka-Volterra system:32

−d1Δu = u(a1 − b1u− c1v) + g1 in Ω,

−d2Δv = v(a2 − b2v − c2u) + g2 in Ω,

u|Γ = 0, v|Γ = 0 on Γ.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (6.51)

This system has applications in ecology:u, v ≥ 0 are the unknown concentrations of two species,a1, a2 are the birth (or, if a’s are negative, death) rates,b1, b2 > 0 are related to the carrying capacities of the environment,c1, c2 are the interaction rates,d1, d2 > 0 are diffusion coefficients related to migrations,g1, g2 ≥ 0 are the outer supply rates.

If both c1 and c2 are positive, (6.51) describes a competition-in-ecology type modelwhile, if both c1 and c2 are negative, (6.51) refers to a cooperation-in-ecology typemodel. Eventually, if c1 > 0 and c2 < 0, we get a predator/prey model; u is thenthe prey species concentration while v is the predator concentration.

The peculiarity of this system is its non-coercivity similarly like in the case ofthe systems in the previous section 6.2 for p < 2. Thus, again we will employ thefixed-point technigue, specifically here by involving the mapping (u, v) �→ (u, v)where (u, v) ∈ W 1,2(Ω)2 is the weak solution to the following two equations:

−d1Δu = u(a1 − b1u+ − c1v) + g1 in Ω,

−d2Δv = v(a2 − b2v+ − c2u) + g2 in Ω,

u|Γ = 0, v|Γ = 0 on Γ.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (6.52)

Existence of a weak solution u and v to these de-coupled equations can be shown,e.g., by a direct method, cf. Proposition 4.16, under assumptions made below. Letus agree to consider ‖v‖W 1,2

0 (Ω) := ‖∇v‖L2(Ω;Rn).

31Hint: Consider the mapping (v, ω) �→ (u, w) with (u,w) solving the system

(v·∇)u− divσ(e(u)

)+∇π = g(1 − αγ(w)), div u = 0, v·∇w − div

(β′(ω)∇w

)= 0 .

Realize that, for any (v, ω), this system is decoupled and partly linearized, and (u,w) is indeeddetermined uniquely. Then use Schauder’s fixed point theorem.

32Original studies of oscillation in biological or ecological systems (not necessarily in the pres-ence of diffusion) originated in Lotka [264] and Volterra [423] and later received intensive scrutiny,cf. e.g. Pao [324, Sect.12.4–6].

6.3. Reaction-diffusion system 187

Lemma 6.22 (Non-negativity of u and v). Let u, v ≥ 0 a.e., and a1 < d1N−2+

c−1 ess supx∈Ωv(x) and a2 < d2N−2 + c−2 ess supx∈Ωu(x) with N the norm of the

embedding W 1,20 (Ω) ⊂ L2(Ω). Then u, v ≥ 0 a.e. in Ω.

Proof. Let us first consider c1 ≥ 0 and test the first equation in (6.52) by u− andnotice that b1u

+u− = 0, which gives

d1‖u−‖2W 1,20 (Ω)

≤∫Ω

d1|∇u−|2 + c1v(u−)2 − g1u

− dx = a1‖u−‖2L2(Ω). (6.53)

If N2a1 < d1, we can absorb the last term in the left-hand side, which thenimmediately gives u− = 0 a.e., so u ≥ 0 a.e. in Ω. For c1 < 0, we must estimate

d1‖u−‖2W 1,20 (Ω)

≤∫Ω

d1|∇u−|2 − g1u− dx =

∫Ω

a1(u−)2

− c1v(u−)2 dx ≤ (a1 + |c1| ‖v‖L∞(Ω)

)‖u−‖2L2(Ω). (6.54)

Analogous considerations work to show v ≥ 0. �Lemma 6.23 (Upper bounds). Let, in addition to the assumptions in Lem-ma 6.22, also g1, g2 ∈ L∞(Ω), b1 > −c−1 and b2 > −c−2 . Then there is a con-stant K sufficiently large such that u, v ∈ [0,K] a.e. in Ω implies u, v ≤ K a.e. inΩ.

Proof. We test the first equation in (6.52) by (u−K)+ and use (1.50), which gives

d1‖(u−K)+‖2W 1,2

0 (Ω)≤∫Ω

d1|∇(u−K)+|2

+(b1u

2 − a1u− g1 + c1vu)(u −K)+ dx = 0. (6.55)

We take K so large that r �→ b1r2 − a1r − g1(x) + c1v(x)r is non-negative on

[K,+∞), namely b1K2 − a1K − ess supx∈Ωg1(x) + c−1 K

2 ≥ 0 and 2b1K − a1 +2c−1 K ≥ 0. Such K does exist whenever b1 + c−1 > 0. Then (6.55) yields u ≤ Ka.e. in Ω. Analogous considerations are for v. �Lemma 6.24 (Continuity of (u, v) �→ (u, v)). Let

a1 < d1N−2 + c−1 K, a2 < d2N

−2 + c−2 K, b1 > −c−1 , b2 > −c−2 (6.56)

with K so large that Lemma 6.23 is in effect. Then the solution (u, v) to (6.52)is unique and the mapping (u, v) �→ (u, v) is weakly continuous as L2(Ω)2 →W 1,2(Ω)2 if both arguments satisfy 0 ≤ u ≤ K and 0 ≤ v ≤ K.

Proof. Uniqueness of the solution to (6.52): consider two solutions u1, u2 ∈W 1,2(Ω) to the first equation in (6.52) and test the difference by u1 − u2 =: u12.It gives

d1‖u12‖2W 1,20 (Ω)

≤∫Ω

d1|∇u12|2 + b1(u1u

+1 −u2u

+2

)u12 + c1vu

212dx = a1

∫Ω

u212dx,


which gives u12 = 0 if a1 < d1N2 and c1 ≥ 0, or if c1 < 0 and a1 − c1K < d1N

−2.Now, consider a sequence {vk}k∈N converging to v weakly in L2(Ω). The cor-

responding solutions uk are bounded in W 1,2(Ω), hence (up to a subsequence) uk

converges to some u weakly in W 1,2(Ω). Using the compact embedding W 1,2(Ω) ⊂L2(Ω) in the integral identity

∫Ω∇uk∇z− uk(a1 − b1u

+k − c1vk)z− g1z dx = 0 for

any z ∈ W 1,2(Ω)∩L∞(Ω), we get that u is the weak solution to the first equationin (6.52). As this solution is unique, even the whole sequence {uk}k∈N convergesto it. The weak continuity of v �→ u : L2(Ω) → W 1,2(Ω) has thus been shown. Themapping u �→ v can be treated analogously. �Proposition 6.25 (Existence of a solution to (6.51)). Let g1, g2 ∈ L∞(Ω),and the birth rates a1 and a2 be small enough and the carrying capacities b1 andb2 be large enough as specified in (6.56) with K sufficiently large as specified inthe proof of Lemma 6.23. Then there is a solution (u, v) to (6.51) such that 0 ≤u(·) ≤ K, 0 ≤ v(·) ≤ K a.e. on Ω.

Proof. We apply the Schauder fixed-point theorem (cf. Exercise 2.55 modified forthe weak* topology) to the mapping (u, v) �→ (u, v) defined by (6.52) on thecompact convex set {(u, v) ∈ L∞(Ω)2; 0 ≤ u(·) ≤ K, 0 ≤ v(·) ≤ K a.e. on Ω}equipped with the weak topology of L2(Ω)2. Note that this set is mapped intoitself if K is taken suitably as mentioned in Lemma 6.24. As the resulting fixedpoint (u, v) is non-negative, it solves the original system (6.51), too. �Remark 6.26. Existence of steady states especially in non-cooperative ecologicalsystems is not automatic hence it is not surprising that Proposition 6.25 worksonly under rather strong data qualification.

Remark 6.27. It should be emphasized that the mere existence of solutions to thesteady-state Lotka-Volterra system (6.51) is only a basic ambition in this analysis.The research in this area focuses on more advanced questions such as multiplicity ofthe solutions, their stability both with respect to data perturbations and whetherthey attract trajectories of the evolution variant of this system, cf. (12.57), etc.

6.4 Thermistor

We will address the steady-state of electric and temperature fields in an isotropichomogeneous electrically conductive medium occupying the domain Ω whose con-ductivity (both electrical and thermal) depends on temperature which is, viceversa, influenced by the produced Joule’s heat. Electrical devices using these ef-fects to link temperature with electrical properties are called thermistors. Anyhow,a filament in each bulb, working with temperatures ranging many hundreds of de-grees, is addressed by the following system, too:

−div(κ(θ)∇θ

)= σ(θ)|∇φ|2 on Ω , (6.57a)

−div(σ(θ)∇φ

)= 0 on Ω , (6.57b)

6.4. Thermistor 189

with the following interpretation:φ is the electrostatic potential,θ is the temperature,σ the electric conductivity (depending on θ),κ the heat conductivity (depending again on θ).

Therefore, (6.57a) is the heat equation, −κ(θ)∇θ denoting the heat flux governedby the Fourier’s law while (6.57b) is the Kirchhoff’s continuity equation for theelectric current j being governed by Ohm’s law j = σ(θ)∇φ. The (specific) powerof the electric current is the scalar product of j with the intensity ∇φ of the electricfield, i.e. the so-called Joule heat j · ∇φ = σ(θ)|∇φ|2 being the source term in theright-hand side of (6.57a).

Of course, the system (6.57) is to be completed by boundary conditions:e.g. the Dirichlet one on ΓD with measn−1(ΓD) > 0 (= electrodes) and zero Neu-mann condition on ΓN = Γ \ ΓD (= an isolated part), i.e.

θ|ΓD = θD, φ|ΓD = φ

Don ΓD ,

∂θ

∂ν=

∂φ

∂ν= 0 on ΓN. (6.58)

Let us note that the right-hand side of (6.57) has higher homogeneity thanits left-hand side, so that again we meet the phenomenon of loss of coercivity of thewhole system. Inspite of this, we get existence of solutions under rather generaldata qualification. The basic trick33 relies on the special feature that the Jouleheat σ(θ)|∇φ|2 with φ ∈ W 1,2(Ω) is not only in L1(Ω) but also in W 1,2(Ω)∗ if(6.57b) holds, and consists in the transformation of (6.57) into the system34

div(κ(θ)∇θ + σ(θ)φ∇φ

)= 0, (6.59a)

div(σ(θ)∇φ

)= 0. (6.59b)

Now, again the crucial point is to design a fixed-point scheme suitably. Here,an advantageous option is to decouple the system (6.59) as follows:

div(κ(ϑ)∇θ + σ(ϑ)φ∇φ

)= 0, (6.60a)

div(σ(ϑ)∇φ

)= 0, (6.60b)

this means we consider the mapping

M := M2 ◦(M1×id

): ϑ �→ θ, M1 : ϑ �→ φ, M2 : (φ, ϑ) �→ θ, (6.61)

where, for ϑ given, φ solves in a weak sense (6.60b) and then θ solves in a weaksense (6.60b), considering naturally the boundary conditions (6.58). The existenceand uniqueness of such solutions has been proved in Chapter 2.

33Without this trick, one must use regularity to guarantee strong convergence of φ’s in W 1,p(Ω)and then |∇φ|2 in a suitable Lp/2(Ω) ⊂ W 1,p(Ω)∗ or, possibly after Kirchhoff’s transformation,the L1-theory like in Remark 6.14 together with the strong convergence as in Exercise 6.33.

34Realize that, by the formula div(av) = adiv v +∇a · v and by (6.57b), one indeed has

div(σ(θ)φ∇φ

)= div

(σ(θ)∇φ

)φ+ σ(θ)∇φ · ∇φ = σ(θ)∇φ · ∇φ = Joule’s heat.


Lemma 6.28 (A-priori estimates). Let us assume σ : R → R and κ : R → Rcontinuous, 0 < cσ ≤ σ(·) ≤ Cσ, 0 < cκ ≤ κ(·) ≤ Cκ, θD

= θ0|Γ, and φD= φ0|Γ

for some θ0, φ0 ∈ W 1,2(Ω) ∩ L∞(Ω), and ϑ ∈ W 1,2(Ω) arbitrary, and let φ and θsolve (6.60)–(6.58) in the weak sense. Then, for some constants C1, C2, and C3

independent of ϑ, ∥∥φ∥∥L∞(Ω)

≤ C1 , (6.62a)∥∥∇φ∥∥L2(Ω;Rn)

≤ C2, (6.62b)∥∥θ∥∥W 1,2(Ω)

≤ C3 . (6.62c)

Proof. The estimate (6.62b) follows by using the test function v = φ − φ0 ∈W 1,2(Ω) in a weak formulation of (6.60b); note that obviously v|ΓD = 0.

The estimate (6.62a) follows by testing (6.60b) by v = (φ ∓ ‖φ0‖L∞(Ω))± as

in Exercise 2.76; note that again v|ΓD = 0.The estimate (6.62c) can be obtained by using the test function v = θ −

θ0 ∈ W 1,20 (Ω) in a weak formulation of (6.60a); note that obviously v|ΓD = 0. By

Holder’s inequality, this leads to the estimate

cκ‖∇θ‖2L2(Ω;Rn) ≤∫Ω

κ(ϑ)∇θ·∇θdx =

∫Ω

κ(ϑ)∇θ·∇θ0 − σ(ϑ)φ∇φ·∇(θ−θ0) dx

≤ ∥∥κ(ϑ)∥∥L∞(Ω)

∥∥∇θ∥∥L2(Ω;Rn)

∥∥∇θ0∥∥L2(Ω;Rn)

+∥∥σ(ϑ)∥∥

L∞(Ω)

∥∥φ∥∥L∞(Ω)

∥∥∇φ∥∥L2(Ω;Rn)

∥∥∇θ −∇θ0∥∥L2(Ω;Rn)

≤ (Cκ

∥∥∇θ0∥∥L2(Ω;Rn)

+CσC1C2

)∥∥∇θ∥∥L2(Ω;Rn)

+ CσC1C2

∥∥∇θ0∥∥L2(Ω;Rn)

. �

Proposition 6.29. The system (6.57)–(6.58) has a weak solution (θ, φ) ∈ W 1,2(Ω)2.

Proof. We take a sufficiently large ball in W 1,2(Ω), namely

B ={θ∈W 1,2(Ω); ‖θ‖W 1,2(Ω) ≤ C3

}, (6.63)

and apply Schauder’s fixed-point Theorem 1.9 (cf. Exercise 2.55) for the mappingM defined by (6.61) on B endowed by the weak topology which makes it compact.For this, we have to prove the weak continuity of the mapping M : ϑ �→ θ :W 1,2(Ω) → W 1,2(Ω). Supposing ϑk ⇀ ϑ weakly in W 1,2(Ω) hence strongly inLp∗−ε(Ω), we get

φk ⇀ φ(weakly in W 1,2(Ω)

)(6.64)

hence strongly in Lp∗−ε(Ω), from which we then get

θk ⇀ θ(weakly in W 1,2(Ω)

). (6.65)

For (6.64), we used σ(ϑk) → σ(ϑ) in any Lq(Ω), q < +∞, and made the limitpassage in the identity

0 =

∫Ω

σ(ϑk)∇φk · ∇v dx →∫Ω

σ(ϑ)∇φ · ∇v dx (6.66)

6.4. Thermistor 191

for v ∈ W 1,∞(Ω) which is a dense subset in W 1,2(Ω). Also, we used uniqueness ofφ solving (6.60b) for ϑ fixed, which is obvious since (6.60b) is a linear equation.Furthermore, for (6.65) we used κ(ϑk) → κ(ϑ) in any Lq(Ω), q < +∞ and instrong convergence φk → φ, cf. (6.64), and made the limit passage in the identity

0 =

∫Ω

κ(ϑk)∇θk · ∇v + σ(ϑk)φk∇φk · ∇v dx

→∫Ω

κ(ϑ)∇θ · ∇v + σ(ϑ)φ∇φ · ∇v dx, (6.67)

which holds for v smooth enough, say W 1,∞(Ω). Again, we use also uniqueness ofθ solving (6.60a) for φ and ϑ fixed.

Then, by the Schauder theorem, M has a fixed point θ ∈ B, and then obvi-ously the couple (θ, φ) with φ = M1(θ) solves (6.57). �

Exercise 6.30 (Other boundary conditions). Assume κ constant, and modify theboundary conditions (6.58) as

κ∂θ

∂ν+ bθ = bθe and σ(θ)

∂φ

∂ν= je on Γ,

with b ≥ 0, and the external temperature θe ∈ L∞(Γ) and the prescribed elec-

tric current je ∈ L2#′(Γ). Show solvability of such a modified system provided∫

Γ jedS = 0 and provided κ is constant.35 Additionally, modify the linear heattransfer by considering the Stefan-Boltzmann conditions, cf. (2.125); this wouldapply, e.g., to a lamp filament working in high temperatures where the heat/lightradiation mechanism intentionally dominates the usual heat convection.

35Hint: For M1, realize that the zero-current condition∫ΓjedS = 0 is necessary to ensure

existence of φ and that only ∇φ is determined uniquely while φ is determined only up toconstants, and then this nonuniqueness is smeared out by M2. Note that (6.60b) has a vari-ational structure of minimizing the convex potential φ �→ ∫

Ω12σ(ϑ)|∇φ|2dx − ∫

Γ jeφ dS on

{v∈W 1,2(Ω);∫Ω φdx = 0}. The zero-current condition is necessary to ensure that this functional

is finite. Show coercivity by estimating∫Ω

1

2σ(ϑ)|∇φ|2dx−

∫ΓjeφdS ≥ 1

2inf σ(·)‖∇φ‖2

L2(Ω)− ‖je‖

L2#′(Γ)‖φ‖

L2#(Γ)

≥ 1

4inf σ(·)C−2

P

(‖φ‖2

W1,2(Ω)− ∣∣ ∫

Ωφ dx

∣∣2)−N‖je‖L2#

′(Γ)‖φ‖W1,2(Ω),

where CP is the constant from the Poincare inequality (1.58) used for p = 2, and N is the norm of

the trace operator u �→ u|Γ : W 1,2(Ω)→ L2#(Γ). Thus we can see that this functional is coerciveon the mentioned subspace of W 1,2(Ω). Even it is strictly convex. As its differential is evenstrongly monotone, the continuity of ϑ → ∇φ follows. As the L∞-estimate of φ cannot now beexpected, proceed directly by using L1-theory for the linear equation div

(κ(ϑ)∇θ

)+σ(ϑ)|∇φ|2 =

0, use a-priori estimates of θ ∈ Wλ,2(Ω) � Ln/(n−2)−ε(Ω) with λ < 2−n/2, cf. Proposition 3.31,and (not relying on uniqueness but only on convexity of the set of solutions for fixed ϑ) employKakutani’s fixed point theorem 1.11.


Exercise 6.31. Prove (6.66) and (6.67) directly for v ∈ W 1,2(Ω).36

Exercise 6.32. Again, from natural reasons, uniqueness of the weak solution to thewhole system (6.59) can be expected only for small data. Prove this uniqueness,assuming σ and κ Lipschitz continuous and both θ

Dand φ

Dare small enough.37

Exercise 6.33. Prove strong convergence in (6.64) and in (6.65).38

6.5 Semiconductors

Semiconductor devices, such as diodes, bipolar and unipolar transistors, thyristors,etc., and their systems in integrated circuits, have formed a technological base offast industrial and post-industrial development of mankind in the 2nd half of the20th century.39 Mathematical modelling of particular semiconductor devices usesvarious models. The basic, so-called drift-diffusion model has been formulated byRoosbroeck [355] and, in the steady-state isothermal variant, is governed by thefollowing system40

div(ε∇φ

)= n− p+ c

Din Ω, (6.68a)

div(∇n− n∇φ

)= r(n, p) in Ω, (6.68b)

div(∇p+ p∇φ

)= r(n, p) in Ω, (6.68c)

36Hint: Consider the Nemytskiı mapping Na determined by the integrand a:Ω×R →Rn:(x, r) �→ σ(r)∇v(x) and, verifying (1.48), show its continuity as Na:Lp∗−ε(Ω)→ L2(Ω;Rn).For (6.67), consider a:(x, r) �→ κ(r)∇v(x).

37Hint: Imitate the strategy of Exercise 6.20. In particular, the term div(σ(θ)φ∇φ) results to∫Ω

(σ(θ1)φ1∇φ1)−σ(θ2)φ2∇φ2)

)·∇θ12 dx

=

∫Ω

(σ(θ1)−σ(θ2)

)φ1∇φ1 + σ(θ2)φ12∇φ1 + σ(θ2)φ2∇φ12

)·∇θ12 dx

and then use Holder inequality to estimate it “on the right-hand side”.38Hint: Use uniform monotonicity and cσ‖∇φk−∇φ‖L2(Ω;Rn) ≤

∫Ωσ(ϑk)|∇φk−∇φ|2dx =∫

Ω σ(ϑk)∇φk ·∇(φk−φ) − σ(ϑk)∇φ·∇(φk−φ) dx = − ∫Ω σ(ϑk)∇φ·∇(φk−φ) dx→ 0. In the case

(6.65), use the weak lower-semicontinuity of (ϑk , θk) �→∫Ω κ(ϑk)|∇θk|2dx:

limk→∞

cκ‖∇θk−∇θ‖L2(Ω;Rn) ≤ lim supk→∞

∫Ωκ(ϑk)|∇θk−∇θ|2dx

= − lim infk→∞

∫Ωκ(ϑk)∇θk·∇(θk−θ) dx− lim

k→∞

∫Ωκ(ϑk)∇θ·∇(θk−θ) dx ≤ 0.

39This was reflected by Nobel prizes awarded for discovery of the transistor effect toW.B. Shockley, J. Bardeen, and W.H. Brattain in 1956, for invention of integrated circuitsto J.S. Kilby in 2000, and for semiconductor heterostructures to Z.I. Alferov and H. Kroemeralso in 2000.

40For more details, the reader is referred to the monographs by Markowich [275], Markowich,Ringhofer, and Schmeiser [276], Mock [289], or Selberherr [380], or to papers, e.g., by Gajewski[162], Groger [191], Jerome [216], or Mock [288]. The model (6.68) can be derived from particle-type models on the assumption that the average distance between two subsequent collisions tendsto zero; cf. [276].

6.5. Semiconductors 193

where we use the conventional notation41

n a concentration of the negative-charge carriers (i.e. of the electrons),p a concentration of the positive-charge carriers (the so-called holes),φ the electrostatic potential,cD=c

D(x) a given profile of concentration of dopants (=donors−acceptors),

ε > 0 a given permitivity,r = r(n, p) generation and recombination rate, cf. Example 6.37 below.

The so-called Poisson equation42 (6.68a) is the rest of Maxwell’s equations whenneglecting magnetic-field effects, which says that divergence of the electric induc-tion ε∇φ has as the source the total electric charge n − p + c

D. The equation

(6.68b) is the continuity equation for the phenomenological electron current43

jn = ∇n − n∇φ with the source r = r(n, p). The equation (6.68c) has a similarmeaning for the phenomenological hole current jp = −∇p− p∇φ.

Of course, (6.68) is to be completed by boundary conditions: let us consider,for simplicity, the Dirichlet one on ΓD with measn−1(ΓD) > 0 (which describesconventional electrodes) and zero Neumann on ΓN = Γ \ ΓD (an isolated part), i.e.

φ|ΓD = φD, n|ΓD = n

D, p|ΓD = p

Don ΓD,

∂φ

∂ν=

∂n

∂ν=

∂p

∂ν= 0 on ΓN. (6.69)

Examples of geometry of typical semiconductor devices, a bi-polar and a uni-polartransistors, are in Figure 15.44

> 0cc

c

< 0= 0

of dopants)

scale:

(concentrationD

D

DΓ0 0Γ00Γ

Γ1

Γ1Γ1ΓΓ0 1Γ

1Γ Ω Ω

base gate drainsourceemitter collectorinsulator

substrate substrate

the surface

of the chip

The grey

Figure 15. Schematic geometry of a bi-polar transistor (left) and a uni-polar field-effecttransistor (so-called FET) (right) which are basic elements of integrated cir-cuits manufactured by an epitaxial technology. The grey scale refers to thelevel of dopants (hence the left figure refers to a so-called p-n-p transistor).

41Hopefully, “n” and “p” used in this section causes no confusion with the dimension n of thedomain Ω ⊂ Rn used also here, or the integrability in Lp(Ω) spaces used in other parts.

42More precisely, the Poisson equation is Δu = g. For g = 0 it is called the Laplace equation.43For simplicity, we consider diffusivity and mobility constant (and equal 1). Dependence

especially on ∇φ is, however, often important and may even create instability of steady-stateson which operational regimes of special devices, so-called Gunn’s diodes, made from binarysemiconductors (e.g. GaAs) are based; such diodes have no steady state under some voltageand therefore must oscillate (typically on very high frequencies ranging GHz). For mathematicalanalysis of such system see Frehse and Naumann [152] or Markowich, Ringhofer, and Schmeiser[276, Sect.4.8].

44Transistors have always three electrodes. In the bi-polar transistor, Figure 15(left), ΓD hastherefore three disjoint components. In the unipolar transistor, Figure 15(right), ΓD has onlytwo components, the third electrode, called a gate, is realized through Newton-type boundaryconditions ε ∂φ

∂ν= (φ−φG) instead of the Neumann one (6.69), with φG denoting the electrostatic

potential of the gate, cf. Exercise 6.38.


A substantial trick consists in a nonlinear transformation: we introduce anew variable set (φ, u, v) related to (φ, n, p) by

n = eφu , p = e−φv , (6.70)

and abbreviate

s(φ, u, v) := r(eφu, e−φv) and σ(φ, u, v) =s(φ, u, v)

uv − 1. (6.71)

Let us remark that −ln(u) and ln(v) are called quasi-Fermi potentials of electronsand holes, respectively. Obviously, (6.70) transforms the currents jn and jp to

jn = ∇n− n∇φ = eφ∇u + eφ∇φu − eφ∇φu = eφ∇u , (6.72a)

jp = −∇p− p∇φ = −e−φ∇v+e−φ∇φv−e−φ∇φ v = −e−φ∇v, (6.72b)

and thus the system (6.68) transforms to

div(ε∇φ) = eφu− e−φv + cD, (6.73a)

div(eφ∇u) = s(φ, u, v), (6.73b)

div(e−φ∇v) = s(φ, u, v), (6.73c)

while the boundary conditions (6.69) transform to

φ|ΓD = φD , u|ΓD = uD := e−φDnD , v|ΓD = vD := eφDpD on ΓD, (6.74a)

∂φ

∂ν=

∂u

∂ν=

∂v

∂ν= 0 on ΓN. (6.74b)

We will again use the fixed-point technique, designed by means of a map-ping M(u, v) = M2(M1(u, v), u, v), where M1 : (u, v) �→ φ=the weak solution to(6.73a) with (6.74), and M2 : (φ, u, v) �→ (u, v)= the weak solutions to:

div(eφ∇u)= σ(φ, u, v)(uv − 1), (6.75a)

div(e−φ∇v)= σ(φ, u, v)(uv − 1), (6.75b)

with the boundary conditions (6.74). We assume φD, n

D, p

D∈ L∞(ΓD), nD

(·) ≥ δ,pD(·) ≥ δ with some δ > 0, so that one can take K ≥ 1 such that uD andvD

are in [e−K , eK ]. Moreover, let φD

= φ0|Γ, uD = u0|Γ, vD= v0|Γ for some

φ0, u0, v0 ∈ W 1,2(Ω) ∩ L∞(Ω).

Lemma 6.34 (A-priori estimates). Let σ : R × R+ × R+ → R be a positivecontinuous function. For u, v ∈ [e−K , eK ], (6.73a) with the boundary conditionfrom (6.74) has a unique weak solution φ ∈ W 1,2(Ω) ∩ L∞(Ω) satisfying

φ(x) ∈ [φmin, φmax] for a.a. x∈Ω, (6.76)


with φmin ∈ R so small and φmax ∈ R so large that

φmin ≤ infx∈ΓD

φD(x) , eφmin+K − e−φmin−K + sup

x∈ΩcD(x) ≤ 0, (6.77a)

φmax ≥ supx∈ΓD

φD(x) , eφmax−K − e−φmax+K+ inf

x∈ΩcD(x) ≥ 0. (6.77b)

Moreover, for φ ∈ L∞(Ω) and u, v ∈ [e−K , eK ], (6.74)–(6.75) have unique weaksolutions u and v satisfying, for some CK depending on K,

‖u‖W 1,2(Ω) ≤ CK , ‖v‖W 1,2(Ω) ≤ CK , (6.78a)

u(x), v(x) ∈ [e−K , eK ] for a.a. x∈Ω. (6.78b)

Proof. Use the direct method for the strictly convex and coercive potential φ �→∫Ω

12ε|∇φ|2+ueφ+ve−φ−c

Dφdx on the affine manifold {φ ∈ W 1,2(Ω); φ|ΓD = φ

D};

note that this functional can take the value +∞. We thus get a unique weak45

solution φ = M1(u, v) to the equation (6.73a) with the boundary condition from(6.74). The W 1,2-estimate can be obtained by a test of (6.73a) by φ−φ0: realizingthat always e−φ(φ − φ0)

+ ≤ e‖φ0‖L∞(Ω) and −eφ(φ − φ0)− ≤ e‖φ0‖L∞(Ω) , we have

by Green’s Theorem 1.31∫Ω

ε|∇φ|2 dx ≤∫Ω

ε|∇φ|2 + eφ(φ− φ0)+u− e−φ(φ − φ0)

−v dx

=

∫Ω

cD(φ0 − φ) − eφ(φ−φ0)−u+ e−φ(φ−φ0)

+v + ε∇φ · ∇φ0 dx

≤ ‖cD‖L∞(Ω)

(‖φ‖L1(Ω) + ‖φ0‖L1(Ω)

)+ measn(Ω)e

K+‖φ0‖L∞(Ω) + ε‖∇φ‖L2(Ω;Rn)‖∇φ0‖L2(Ω;Rn),

from which an a-priori bound for φ in W 1,2(Ω) follows. The upper bound in (6.76)can be shown by a comparison likewise in Exercise 2.76, here we use the testfunction z := (φ−φmax)

+. Note that the first condition in (6.77b) implies z|ΓD = 0hence it is indeed a legal test function for the weakly formulated boundary-valueproblem (6.73a)-(6.74). This test gives∫

Ω

ε∇φ · ∇(φ − φmax)+ +

(eφu− e−φv + c

D

)(φ− φmax)

+ dx = 0. (6.79)

Now, we realize that the first term in (6.79) is always non-negative, cf. (1.50), andthat, if u ≥ e−K and v ≤ eK , then necessarily eφu − e−φv + c

D> 0 wherever

(φ − φmax)+ > 0 with φmax satisfying the second inequality in (6.77b). We can

therefore see that (6.79) yields (φ − φmax)+ ≤ 0 a.e. in Ω. The lower bound in

(6.76) can be shown similarly by testing (6.73a) by z := (φ− φmin)−.

45This sort of solution is called a variational solution. If, however, we show a-posteriori bound-edness in L∞(Ω), cf. (6.76), this solution is the weak solution. One can also imagine the monotonenonlinearity r �→ u(x)er−v(x)e−r in (6.73a) modified, for a moment, out of [φmin, φmax] to havea subcritical polynomial growth.


The unique weak solution u to the linear boundary-value problem (6.75a)–(6.74) obviously does exist. The a-priori estimate can be obtained by testing(6.75a) by u− u0:

eφmin

∫Ω

|∇u|2dx ≤∫Ω

eφ|∇u|2 + σ(φ, u, v)u2v dx

=

∫Ω

eφ∇u · ∇u0 + σ(φ, u, v)(uu0v + u− u0) dx

≤ eφmax‖∇u‖L2(Ω;Rn)‖∇u0‖L2(Ω;Rn)

+ Cσ

(‖u‖L2(Ω)‖u0‖L2(Ω)e

K+ ‖u‖L1(Ω) + ‖u0‖L1(Ω)

)(6.80)

where Cσ := sup[φmin,φmax]×[e−K,eK ]2 σ(·, ·, ·) so that u is bounded in W 1,2(Ω). Theupper bound for u in (6.78b) can be shown again by a comparison, now by choosingz := (u− eK)+ as a test function for (6.75a). As u|ΓD = uD ≤ eK due to the choiceof K, z|ΓD = 0 hence it is indeed a legal test function for the weakly formulatedboundary-value problem (6.75a)-(6.74). This test gives∫

Ω

eφ∇u · ∇(u− eK)+ + σ(φ, u, v)(uv − 1)(u − eK)+ dx = 0. (6.81)

As in (6.79), the first term in (6.81) is always non-negative and, if v ≥ e−K , thesecond term is positive wherever u− eK > 0, and we can therefore see that (6.81)yields u ≤ eK a.e. in Ω. The lower bound in (6.78b) can be proved similarly bytesting (6.75a) by z := (u− e−K)−.

Analogous considerations hold for v. �

Lemma 6.35 (Continuity). Let σ : R× R+ × R+ → R+ be continuous.

(i) The mapping M1 : (u, v) �→ φ : L2(Ω)2 → W 1,2(Ω) is weakly continuous ifrestricted on {(u, v); (6.78b) holds}.

(ii) The mapping M2 : (φ, u, v) �→ (u, v) : L2(Ω)3 → W 1,2(Ω)2 is demicontinuousif restricted on {(φ, u, v); (6.76) and (6.78b) hold}.

Proof. Assume uk ⇀ u and vk ⇀ v in L2(Ω). Then, consider φk = M1(uk, vk) and(possibly for a subsequence) φk ⇀ φ in W 1,2(Ω). Then φk → φ in L2∗−ε(Ω), andalso eφkuk ⇀ eφu and eφkvk ⇀ eφv in L2(Ω) provided φk is bounded in L∞(Ω),as it really is due to (6.76).46 Then one can pass to the limit in the identity∫Ω ε∇φk · ∇z + eφkukz − eφkvkz + cDz dx = 0, showing that φ = M1(u, v) and, infact, the whole sequence converges.

As to (ii), considering (φk, uk, vk) → (φ, u, v) in L2(Ω)3, by the a-priori es-timate (6.78a) the corresponding sequence (uk, vk) converges (at least as a sub-sequence) weakly in W 1,2(Ω)2 to some (u, v). Passing to the limit in the integral

46Realize that we can imagine that the nonlinearities (ξ, r) �→ eξr and (ξ, r) �→ e−ξr aremodified for ξ �∈ [φmin, φmax] to have a linear growth.


identities47 ∫Ω

eφk∇uk · ∇z + σ(φk, uk, vk)(ukvk−1)z dx = 0, (6.82a)∫Ω

e−φk∇vk · ∇z + σ(φk, uk, vk)(ukvk−1)z dx = 0 (6.82b)

for all z ∈ W 1,∞(Ω), z|ΓD = 0, we can see that (u, v) solves (6.75) with theboundary conditions (6.74). As σ ≥ 0 and also u ≥ 0 and v ≥ 0, this (u, v) mustbe unique and thus the whole sequence converges to it. �

Proposition 6.36 (Existence). Under the above assumptions on σ, cD, n

D, p

D,

and φD, the system (6.73)–(6.74) has a weak solution.

Proof. Use Schauder fixed-point Theorem 1.9 for the mapping M = M2 ◦M1 onS := {(u, v) ∈ L∞(Ω)×L∞(Ω); (6.78b) holds} equipped with the norm topologyof L2(Ω). Realize that, by (6.78a) and Rellich-Kondrachov’s Theorem 1.21, M(S)is indeed relatively compact. �

Example 6.37 (Shockley-Read-Hall model). The generation/recombination rate isoften modelled by

r = r(n, p) =np− c2int

τn(n+ cint) + τp(p+ cint)(6.83)

with cint > 0 an intrinsic concentration and τn > 0 and τp > 0 the electron andthe hole live-time, respectively. Assuming, without loss of generality if suitablephysical units are chosen, that cint = 1, the model (6.83) indeed gives σ as apositive continuous function as required in Lemma 6.34, namely

σ(φ, u, v) =1

τn(eφu+ 1) + τp(e−φv + 1). (6.84)

Exercise 6.38 (Newton boundary conditions for φ). Modify Lemma 6.34 for com-bining the boundary Dirichlet/Neumann boundary conditions (6.69) with theNewton one: ε∂φ

∂ν = (φ − φG) on some part of ΓN with φG ∈ L∞(ΓN); this part ofΓN corresponds to the so-called gate of an FET-transistor on Figure 15(right).

Exercise 6.39. Strengthen Lemma 6.35 by proving the total continuity of M1 andthe continuity of M2.

48

47As ∇z ∈ L∞(Ω;Rn), we can use eφk → eφ and e−φk → e−φ in L2(Ω) if (6.76) holds. By(6.78b), we can assume σ(φk , uk, vk)→ σ(φ, u, v) in any Lr(Ω), r < +∞, which allows us to passto the limit in the last terms in (6.82a,b). Eventually, the resulting identities can be extendedfor z from W 1,∞(Ω) onto the whole W 1,2(Ω).

48Hint: Show φk → φ in W 1,2(Ω) due to the strong monotonicity of the Laplacean with theDirichlet boundary condition on ΓD by testing the difference of weak formulations determining


Remark 6.40 (Uniqueness). The weak solution to (6.68)–(6.69), whose existencewas proved in Proposition 6.36, is unique only on special occasions. In general,there are even semiconductor devices such as thyristors whose operational regimesjust exploit non-uniqueness of steady states.

respectively φk and φ by φk − φ, which gives∫Ωε∣∣∇φk−∇φ

∣∣2dx =

∫Ω

(eφkuk − eφu− e−φkvk + e−φv

)(φk−φ) dx→ 0.

As to uk, use the uniform (with respect to k) strong monotonicity of u �→ −div(eφk∇u) likewisein Exercise 2.75, and test (6.82a) by z := uk − u:

eφmin∥∥∇(uk−u)

∥∥2L2(Ω;Rn)

≤∫Ωeφk

∣∣∇(uk−u)∣∣2dx

=

∫Ωσ(φk , uk, vk)(uk vk−1)(uk−u)− eφk∇u · ∇(uk−u) dx→ 0.

Eventually, vk → v is similar.

Part II

EVOLUTION PROBLEMS

Chapter 7

Special auxiliary tools

In evolution problems, one scalar variable, denoted by t and having a meaningof time, takes a special role, which is also reflected by mathematical analysis. Inparticular, here we first present a few useful assertions about spaces of abstractfunctions on a “time” interval I := [0, T ], introduced already in Section 1.5, butnow possessing additionally derivatives with respect to time. Always, T will denotea fixed finite time horizon.1

7.1 Sobolev-Bochner space W 1,p,q(I; V1, V2)

For V1 a Banach space and V2 a locally convex space, V1 ⊂ V2, let us define

W 1,p,q(I;V1, V2) :={u ∈ Lp(I;V1);

du

dt∈ Lq(I;V2)

}(7.1)

with ddtu denoting the distributional derivative of u understood as the abstract

linear operator ddtu ∈ L (D(I), (V2,weak)) defined by

du

dt(ϕ) = −

∫ T

0

udϕ

dtdt (7.2)

for any ϕ ∈ D(I), where D(I) stands for infinitely differentiable functions witha compact support in (0, T ). Mostly, both V1 and V2 will be Banach spaces,and then W 1,p,q(I;V1, V2) itself is a Banach space if equipped with the norm‖u‖W 1,p,q(I;V1,V2) := ‖u‖Lp(I;V1) + ‖ d

dtu‖Lq(I;V2). Sometimes, V1 = V2 will occurand then we will briefly write

W 1,p(I;V ) := W 1,∞,p(I;V, V ). (7.3)

1For more detailed study, the reader is referred e.g. to monographs by Gajewski at al. [168,Sect.IV.1] or Zeidler [427, Chap.23].

T. Roubí ek, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics 153, DOI 10.1007/978-3-0348-0513-1_7, © Springer Basel 2013

201

202 Chapter 7. Special auxiliary tools

Occasionally, we use also spaces having 2nd-order time derivatives valued in V3:

W 2,∞,p,q(I;V1, V2, V3) :={u ∈ L∞(I;V1);

du

dt∈Lp(I;V2) and

d2u

dt2∈Lq(I;V3)

}. (7.4)

As to V2 in (7.1), certain degrees of generality will be found useful forthe Rothe and the Galerkin method below, namely replacement of Lq(I;V2) byM (I;V2) or considering V2 a metrizable locally convex space, respectively. As tothe former generalization, we just replace Lp with M in (7.1) and equip it withthe norm counting the total variation of u(·) in V2, i.e. ‖u‖Lp(I;V1)+‖ d

dtu‖M (I;V2);cf. (7.40) below. As to the latter generalization, without loss of generality, wecan assume the topology of V2 generated by a countable collection of seminorms{| · |�}�∈N. Then (7.1) defines a locally convex space of functions u ∈ Lp(I;V1) suchthat ∣∣∣du

dt

∣∣∣q,�

:=

(∫ T

0

∣∣∣dudt

∣∣∣q�dt

)1/q

< +∞ (7.5)

for any ∈N; we then consider W 1,p,q(I;V1, V2) equipped with the topology gen-erated by the functionals u �→ ‖u‖Lp(I;V1) + | ddtu|q,�, ∈ N.

Lemma 7.1. Let p, q ≥ 1 and let V1 ⊂ V2 continuously. Then W 1,p,q(I;V1, V2) ⊂C(I;V2) continuously.

Proof. Let us confine ourselves on V2 a Banach space, the generalization for alocally convex space being clear. Let u ∈ W 1,p,q(I;V1, V2). Then

ddtu is integrable,

and we can put v(t) :=∫ t

0dudϑdϑ. Then∥∥v(t2)− v(t1)∥∥V2

=∥∥∥ ∫ t2

t1

du

dtdt∥∥∥V2

≤∫ t2

t1

∥∥∥dudt

∥∥∥V2

dt. (7.6)

This shows that t �→ v(t) : I → V2 is continuous. Yet, v = u + c, c∈ V2 becauseddtv = d

dtu. Thus u is continuous, too. Moreover, we can estimate

‖v(t)‖V2 ≤∫ T

0

∥∥∥dudt

∥∥∥V2

dt =∥∥∥dudt

∥∥∥L1(I;V2)

≤ Nq

∥∥∥dudt

∥∥∥Lq(I;V2)

, (7.7)

and then

∥∥c∥∥V2

= T−1/p

(∫ T

0

∥∥c∥∥pV2dt

)1/p

= T−1/p∥∥u− v

∥∥Lp(I;V2)

≤ T−1/p∥∥u∥∥

Lp(I;V2)+ T−1/p

∥∥v∥∥Lp(I;V2)

≤ T−1/pN12

∥∥u∥∥Lp(I;V1)

+ T−1/pT 1/pNq

∥∥∥dudt

∥∥∥Lq(I;V2)

, (7.8)

7.1. Sobolev-Bochner space W 1,p,q(I;V1, V2) 203

where Nq and N12 are the norms of the embeddings Lq(0, T ) ⊂ L1(0, T ) andV1 ⊂ V2, respectively.

2 From this, we get

‖u‖C(I;V2) = supt∈I

∥∥∥ ∫ t

0

du

dϑdϑ− c

∥∥∥V2

≤∫ T

0

∥∥∥dudϑ

∥∥∥V2

dϑ+∥∥c∥∥

V2

≤ Nq

∥∥∥dudt

∥∥∥Lq(I;V2)

+ T−1/pN12

∥∥u∥∥Lp(I;V1)

+Nq

∥∥∥dudt

∥∥∥Lq(I;V2)

≤ max(T−1/pN12, 2Nq

)∥∥u∥∥W 1,p,q(I;V1,V2)

. (7.9)

�Lemma 7.2. Let p, q ≥ 1 and let V1 ⊂ V2 continuously. Then C1(I;V1) is containeddensely in W 1,p,q(I;V1, V2).

Proof. Take u ∈ W 1,p,q(I;V1, V2) and, for ε > 0, put

uε(t) :=

∫ T

0

�ε(t+ ξε(t)− s

)u(s) ds, ξε(t) := ε

T − 2t

T, (7.10)

where �ε : R → R is a positive, C∞-function supported on [−ε, ε] and satisfying∫R�ε(t) dt = 1. Such functions are called mollifiers. To be more specific, we can

take

�ε(t) :=

{c ε−1et

2/(t2−ε2) for |t| < ε,0 elsewhere,

(7.11)

with c a suitable constant so that∫R�1(t) dt = 1. Note that the function ξε

(converging to 0 for ε → 0) is just to shift slightly the kernel in the convolutionintegral in (7.10) so that only values of u inside [0, T ] are taken into account,cf. Figure 16.

ε = 1/40

ε = 1/20

ε = 1/10

t = 0 t = Tt = 12T

TTT 000

Figure 16. Example of the “mollifying” kernel s �→ ε(t+ ξε(t)− s

)in the convo-

lutory integral (7.10) for three values t = 0, T/2, and T , and for threevalues ε = T/10, T/20, and T/40.

Denoting �′ε the derivative of �ε, we can write the formula

duε

dt=(1− 2ε

T

)∫ T

0

�′ε(t+ ξε(t)− s

)u(s) ds

=T − 2ε

T

∫ T

0

�ε(t+ ξε(t)− s

)duds

(s) ds. (7.12)

2It holds that Nq = ‖1‖Lq′ (I) = T 1−1/q , cf. also Exercise 2.68.


In particular, the first equality in (7.12) shows that uε∈C1(I;V1). We can estimate∫ T

0

∥∥uε(t)− u(t)∥∥pV1dt =

∫ T

0

∥∥∥∥ ∫ t+ε+ξε(t)

t−ε+ξε(t)

�ε(t+ξε(t)−s

)(u(s)−u(t)

)ds

∥∥∥∥pV1

dt

≤∫ T

0

(∫ ξε(t)+ε

ξε(t)−ε

�ε(h− ξε(t)

)∥∥u(t+h)− u(t)∥∥V1dh

)p

dt

≤(∫ ε

−ε

�ε(h)p′dh

)p−1 ∫ T

0

∫ ξε(t)+ε

ξε(t)−ε

∥∥u(t+h)− u(t)∥∥pV1dhdt

≤ 2p−1cp

ε

∫ T

0

∫ ξε(t)+ε

ξε(t)−ε

∥∥u(t+h)−u(t)∥∥pV1dhdt ≤ 2pcp sup

|h|≤ε

∫ T

0

∥∥u(t+h)−u(t)∥∥pV1dt.

Then we use limε→0 sup|h|≤ε ‖u(·+h)−u‖pLp(I;V1)which is easy to see for u piece-

wise constant while the general case follows by using additionally Proposition 1.36uniformly for the collection {u(·+ h)}|h|≤ε.

3

Analogously, one can show that the last integral in (7.12) approaches ddtu

in Lq(I;V2). Yet, (7.12) says that this integral equals just to ddtuε up to a factor

T/(T − 2ε) converging to 1 when ε → 0. Hence even ddtuε itself converges to d

dtuin Lq(I;V2). �

7.2 Gelfand triple, embedding W 1,p,p′(I;V ,V ∗)⊂C(I;H)

A basic abstract setting for evolution problems relies on the following construction.Let H be a Hilbert space identified with its own dual, H ≡ H∗, and the embeddingV ⊂ H be continuous and dense. Note that then H ⊂ V ∗ continuously; indeed,the adjoint mapping i∗ (which is continuous) to the embedding i : V → H mapsH∗ ≡ H into V ∗ and is injective, i.e.4

u1 �= u2 =⇒ i∗u1 �= i∗u2 ⇐⇒ ∃v ∈ V : 〈u1, v〉 �= 〈u2, v〉. (7.13)

Let us agree to identify i∗u with u if u ∈ H . Thus we may indeed consider H ⊂ V ∗

and the duality pairing between V ∗ and V as a continuous extension of the innerproduct on H , denoted by (·, ·), i.e. for u ∈ H and v ∈ V we have5(

u, v)=⟨u, v

⟩H∗×H

=⟨u, iv

⟩H∗×H

=⟨i∗u, v

⟩V ∗×V

=⟨u, v

⟩V ∗×V

. (7.14)

The indices in (7.14) indicate the spaces paired by the duality. The triple V ⊂H ⊂ V ∗ is called an evolution triple, or sometimes Gelfand’s triple, and the Hilbert

3See e.g. Gajewski et al. [168, Chap.IV, Lemma 1.5].4The equivalence in (7.13) just expresses that the functionals u1 and u2 on H must have

different traces (=restrictions) on any dense subset of H, in particular on V .5The equalities in (7.14) follow subsequently from the identification of H with H∗, the em-

bedding V⊂H, the definition of the adjoint operator i∗, and the identification of i∗u with u.

7.2. Gelfand triple, embedding W 1,p,p′(I;V ,V ∗)⊂C(I;H) 205

space H a pivot. Moreover, the embedding H ⊂ V ∗ is dense. Occasionally, we willneed V or H separable, hence let us assume it generally without restriction ofapplicability to partial differential equations.

Lemma 7.3 (By-part integration formula). Let V ⊂ H ∼= H∗ ⊂ V ∗, andp′ = p/(p−1) be the conjugate exponent to p, cf. (1.20). Then W 1,p,p′

(I;V, V ∗) ⊂C(I;H) continuously and the following by-part integration formula holds for anyu, v ∈ W 1,p,p′

(I;V, V ∗) and any 0 ≤ t1 ≤ t2 ≤ T :

(u(t2), v(t2)

)− (u(t1), v(t1)) = ∫ t2

t1

⟨dudt

, v(t)⟩V ∗×V

+⟨u(t),

dv

dt

⟩V×V ∗

dt. (7.15)

Proof.6 Note that (7.15) holds for u, v ∈ C1(I;V ) by classical calculus, by usingddt (u, v) = (dudt , v) + (u, dv

dt ) = 〈dudt , v〉V ∗×V + 〈u, dvdt 〉V×V ∗ (here (7.14) have been

employed) and integrating it over [t1, t2].Put q = min(2, p). For u ∈ C1(I;V ), we can use (7.15) with v := u, t2 := t

and t1 such that ‖u(t1)‖qH = 1T

∫ T

0 ‖u(ϑ)‖qHdϑ, i.e. the mean value. Thus we get

‖u(t)‖qH = ‖u(t1)‖qH +(‖u(t)‖qH − ‖u(t1)‖qH

)≤ 1

T

∫ T

0

‖u(ϑ)‖qHdϑ+∣∣∣‖u(t)‖2H − ‖u(t1)‖2H

∣∣∣q/2=

1

T

∫ T

0

‖u(ϑ)‖qHdϑ+

∣∣∣∣2 ∫ t

t1

⟨dudϑ

, u(ϑ)⟩dϑ

∣∣∣∣q/2≤ 1

T‖u‖qLq(I;H) + 2q/2

(∥∥∥dudt

∥∥∥Lp′(I;V ∗)

‖u‖Lp(I;V )

)q/2=

1

T‖u‖qLq(I;H) +

∥∥∥dudt

∥∥∥qLp′(I;V ∗)

+ ‖u‖qLp(I;V ), (7.16)

where we used, besides Holder’s inequality, also the inequalities aq − bq ≤ |a2 −b2|q/2, which holds for a, b ≥ 0 and q ∈ [1, 2],7 and (a+ b)q/2 ≤ aq/2 + bq/2. Thenwe use still the estimate

‖u‖Lq(I;H) ≤{

N1‖u‖Lp(I;V ) if p < 2,

N1N2‖u‖Lp(I;V ) if p ≥ 2,(7.17)

where N1 and N2 are the norms of the embedding V ⊂ H and Lp(I) ⊂ L2(I),respectively. As the estimate (7.16) is uniform with respect to t, the continuity

6This proof generalizes that one by Renardy and Rogers [349, p.380] for p �= 2. For p general,see e.g. Gajewski [168, Sect. IV.1.5] or Zeidler [427, Proposition 23.23] where a bit differenttechnique was used.

7This can be proved simply by analyzing the function (1 − ξq)2/|1− ξ2|q with ξ = a/b > 0.This function is either constant=1 for q = 2 or, if q < 2, decreasing on [0,1] and increasing on[1,+∞) and always below 1 (except for ξ = 0 where it equals 1).


of the embedding W 1,p,q(I;V, V ∗) ⊂ C(I;H) has been proved if one confines tofunctions from C1(I;V ).

Yet, the desired embedding as well as the formula (7.15) can be obtained bythe density argument for all functions from W 1,p,q(I;V, V ∗); cf. Lemma 7.2. Thefact that u : I → H is continuous follows from (7.15): if used by v constant andletting t2 → t1, we get (u(t2), v) → (u(t1), v), hence u(·) is weakly continuous, andby v = u we get ‖u(t2)‖H → ‖u(t1)‖H , hence by Theorem 1.2 u(t2) → u(t1) inthe norm topology of H . �

The following approximation property will occasionally be used.

Lemma 7.4. Let 1 ≤ p < +∞. For any u ∈ Lp(I;V )∩L∞(I;H) and any u0 ∈ H,there is a sequence {uε}ε>0 ⊂ W 1,∞,∞(I;V,H) such that

u = limε→0

uε in Lp(I;V ), (7.18a)

lim supε→0

∫ T

0

(duε

dt, uε − u

)dt ≤ 0, (7.18b)

u0 = limε→0

uε(0) in H, (7.18c)

‖uε‖L∞(I;H) ≤ ‖u‖L∞(I;H), u(t) = w-limε→0

uε(t) in H for a.a. t∈I. (7.18d)

Proof. 8 As V ⊂ H densely, we can take {u0ε}ε>0 ⊂ V such that limε→0 u0ε = u0

in H . Then we make the prolongation of u by u0ε for t < 0, let us denote it by uε,and define

uε(t) :=1

ε

∫ +∞

0

e−s/εuε(t−s) ds. (7.19)

In other words, uε is a convolution of uε with the kernel �ε(t) := χ[0,+∞)ε−1e−t/ε.

A simple calculation gives uε(0) = ε−1u0ε

∫ +∞0

e−s/ε ds = u0ε hencefore (7.18c)is proved. Also, {uε(t)}ε>0 is bounded in H and thus converges weakly, for t∈ Ifixed, as a subsequence to some u(t). Simultaneously, for u∗ ∈H , the whole se-quence {〈u∗, uε(t)〉}ε>0 converges to {〈u∗, u(t)〉}ε>0 at each left Lebesgue point of〈u∗, u(·)〉. Using separability of H , we get that u(t) = u(t) for a.a. t∈I, i.e. (7.18d).Also,

∥∥uε

∥∥L∞(I;V )

= maxt∈I

∥∥∥1ε

∫ t

0

e−s/εu(t−s) ds+1

ε

∫ +∞

t

e−s/εu0ε(t−s) ds∥∥∥V

≤ ∥∥ρε∥∥L∞(R)

∥∥u∥∥L1(I;V )

+e−t/ε∥∥u0ε

∥∥V≤ T 1−1/p

ε

∥∥u∥∥Lp(I;V )

+∥∥u0ε

∥∥V.

8Cf. Showalter [383, Sect.III.7]. For p ≥ 2 see also Lions [261, Ch.II, Sect.9.2].

7.3. Aubin-Lions lemma 207

Moreover, on I, it holds

duε

dt=

d

dt

1

ε

∫ +∞

0

e−s/εuε(t−s) ds =d

dt

1

ε

∫ t

−∞e(ξ−t)/εuε(ξ) dξ

=u(t)

ε− 1

ε2

∫ t

−∞e(ξ−t)/εuε(ξ) dξ =

u(t)

ε− 1

ε2

∫ +∞

0

e−s/εuε(t−s) ds =u−uε

ε

(7.20)

where the substitution t − s = ξ has been used twice. In particular, ddtuε ∈

L∞(I;H), and one can test (7.20) by uε − u, which gives∫ T

0

(duε

dt, uε − u

)dt = −1

ε

∫ T

0

‖uε − u‖2H dt ≤ 0 (7.21)

and therefore (7.18b) is certainly valid. Eventually, (7.18a) can be obtained by thecalculations in Lemma 7.2 modified for the kernel �ε specified here. �Remark 7.5. The formula (7.15) for u = v gives

1

2‖u(t2)‖2H − 1

2‖u(t1)‖2H =

∫ t2

t1

⟨dudt

, u(t)⟩V ∗×V

dt . (7.22)

From this formula, one can also see that the function t �→ 12‖u(t)‖2H is absolutely

continuous. Hence, its derivative exists a.e. on I and the chain rule holds:

1

2

d

dt‖u(t)‖2H =

⟨dudt

, u(t)⟩V ∗×V

for a.a. t∈I. (7.23)

7.3 Aubin-Lions lemma

We saw already in Part I that limit passage in lower-order terms needs compactnessarguments. This will be the application of the results presented in this section. Letus first prove one auxiliary inequality which is sometimes referred to as Ehrling’slemma9 if V3 is a Banach space. Here, however, we admit V3 a locally convex space,which will simplify application to the Galerkin method in Section 8.4.

Lemma 7.6 (Ehrling [134], generalized). Let V1, V2 be Banach spaces, andV3 be a metrizable Hausdorff locally convex space, V1 � V2 (compact embedding),and V2 ⊂ V3 (continuous embedding). Then, for any p ≥ 1,

∀ε > 0 ∃a > 0 ∃K∈N ∀v∈V1 : ‖v‖pV2≤ ε‖v‖pV1

+ a

K∑�=1

|v|p� . (7.24)

9The Ehrling lemma says: if V1, V2, V3 are Banach spaces, a linear operator A : V1 → V2

is compact and a linear operator B : V2 → V3 is injective. Then ∀ε > 0 ∃C < +∞ ∀u ∈ V1:‖Au‖V2

≤ ε‖u‖V1+ C‖BAu‖V3

; cf. e.g. Alt [10, p.335]. In the original paper, Ehrling [134]formulated this sort of assertion in less generality.


Proof. Suppose the contrary. Thus we get ε > 0 such that for all a > 0 andK ∈ N there is va,K ∈ V1: ‖va,K‖pV2

> ε‖va,K‖pV1+a

∑K�=1 |va,K |p� . Putting wa,K =

va,K/‖va,K‖V1 , we get

‖wa,K‖pV2≥ ε+ a

K∑�=1

|wa,K |p� (7.25)

and also ‖wa,K‖V2 ≤ N12, the norm of the embedding V1 ⊂ V2. From (7.25)

we get (∑K

�=1 |wa,K |p� )1/p ≤ a−1/pN12 and therefore also |wa,K |� ≤ a−1/pN12 forany a and any K ≥ , and thus lima,K→+∞ |wa,K |� = 0 for any ∈ N. As{wa,K}a>0,K∈N is bounded in V1 and the embedding V1 ⊂ V2 is compact, we have(up to a subsequence) wa,K → w in V2 if a,K → +∞. Hence also |wa,K −w|� → 0for any ∈ N because V2 ⊂ V3 continuous. Clearly,

|w|� ≤ |wa,K |� + |wa,K − w|� → 0

so that |w|� = 0. Hence w = 0 because V3 is assumed a Hausdorff space, so thatwa,K → 0 in V2, which contradicts (7.25). Thus (7.24) is proved. �Lemma 7.7 (Aubin and Lions, generalized10). Let V1, V2 be Banach spaces,and V3 be a metrizable Hausdorff locally convex space, V1 be separable and reflexive,V1 � V2 (a compact embedding), V2 ⊂ V3 (a continuous embedding), 1 < p < +∞,1 ≤ q ≤ +∞. Then

W 1,p,q(I;V1, V3) � Lp(I;V2) (a compact embedding). (7.26)

Proof. We will consider V3 equipped with a collection of seminorm {| · |�}�∈N.We are to prove that bounded sets in W 1,p,q(I;V1, V3) are relatively compact

in Lp(I;V2). Take {uk} a bounded sequence in W 1,p,q(I;V1, V3).11 In particular,

as V1 is reflexive and separable and p ∈ (1,+∞), the Bochner space Lp(I;V1) isreflexive, cf. Proposition 1.38, and thus we have (up to a subsequence)

uk ⇀ u in Lp(I;V1). (7.27)

As always Lq(I;V3) ⊂ L1(I;V3), we have{duk

dt

}k∈N

bounded in L1(I;V3). (7.28)

We may consider u = 0 in (7.27) without loss of generality. Putting v := uk(t)into (7.24) and integrating it over I, we get

‖uk‖pLp(I;V2)≤ ε‖uk‖pLp(I;V1)

+ a

K∑�=1

|uk|pp,� (7.29)

10For the original version with V3 a Banach space see Aubin [25] and Lions [261]. For ageneralization, see also Dubinskiı [127] and Simon [386]. For a nonmetrizable V3, see [359].

11As we address compactness in a Banach space Lp(I;V2), we can work in terms of sequencesonly, which agrees with our definition of compactness of sets as a “sequential” compactness.


where |u|p,� := (∫ T

0 |u(t)|p�dt)1/p, cf. (7.5). The first right-hand-side term canbe made arbitrarily small by taking ε > 0 small independently of k becausesupk∈N ‖uk‖Lp(I;V1) < +∞. Hence, take ε > 0 fixed, which then fixes alsoa and K. Then, for arbitrary but fixed, we are to push to zero the term

|uk|pp,� =∫ T/2

0 |uk(t)|p�dt +∫ T

T/2 |uk(t)|p�dt and we may investigate only, say, the

first integral. Take δ > 0, we can assume δ ≤ T/2. For t ∈ I/2 := [0, T/2] we candecompose

uk = uk + zk , with uk(t) :=1

δ

∫ δ

0

uk(t+ ϑ)dϑ , (7.30)

i.e. uk, being absolutely continuous, represents the “mollified uk”. Thus, usingby-part integration, we have the formula for the remaining zk:

zk(t) =

∫ δ

0

(ϑδ− 1

) d

dϑuk(t+ϑ)dϑ

=

[(ϑδ− 1

)uk(t+ϑ)

]δϑ=0

−∫ δ

0

1

δuk(t+ ϑ)dϑ = uk(t)− uk(t) . (7.31)

Then∫ T/2

0

|uk(t)|p�dt ≤ 2p−1

∫ T/2

0

|uk(t)|p�dt+ 2p−1

∫ T/2

0

|zk(t)|p�dt =: I1,� + I2,�. (7.32)

We can estimate

I2,� ≤∫ T/2

0

(∫ δ

0

(1− ϑ

δ

)∣∣∣ ddt

uk(t+ϑ)∣∣∣�dϑ

)p

dt =

∥∥∥∥∣∣∣duk

dt

∣∣∣�� ψδ

∥∥∥∥pLp(I/2)

=: Ip3,�,

(7.33)where “�” denotes the convolution and ψδ(t) := (t/δ + 1)χ[−δ,0](t). The followingestimate can be proved12:

‖f � g‖Lp(R) ≤ ‖f‖L1(R)‖g‖Lp(R). (7.34)

For f = | ddtuk|� and g = ψδ, we get

I3,� ≤∥∥∥∥∣∣∣duk

dt

∣∣∣�

∥∥∥∥L1(I/2+δ)

∥∥ψδ

∥∥Lp(R)

. (7.35)

By (7.28) and by ‖ψδ‖Lp(R) ≤ p√δ, we have I3,� = O( p

√δ) hence I2,� = O(δ). In

particular, I2,� can be made arbitrarily small if δ > 0 is small enough.

12One can use the trivial estimates ‖f � g‖L1(R) ≤ ‖f‖L1(R)‖g‖L1(R) and ‖f � g‖L∞(R) ≤‖f‖L1(R)‖g‖L∞(R) and, as g �→ f � g is a linear operator, obtain (7.34) by interpolation by theclassical Riesz-Thorin convexity theorem.


Let us now take δ > 0 fixed. By (7.27) with u = 0 and by the definition (7.30)of u, we have uk(t) ⇀ 0 in V1 for every t, hence also uk(t) → 0 in V2 because of thecompactness of the embedding V1 ⊂ V2. Then also |uk(t)|� → 0 because V2 ⊂ V3

is continuous. As {uk}k∈N is bounded in Lp(I;V1), it is bounded in L1(I;V2) too.Thus

∣∣uk(t)∣∣�≤ C1,�‖uk(t)‖V1 ≤ C1,�

δ

∫ δ

0

‖uk(t+ϑ)‖V1dϑ ≤ C1,�T1−1/p

δ‖uk‖Lp(I;V1),

where C1,� = supv |v|�/‖v‖V1 is finite because the embedding V1 ⊂ V3 is assumedcontinuous. Thus uk(t) is bounded in V3 independently of k and t. By Lebesgue

dominated convergence Theorem 1.14, I1,� :=∫ T/2

0|uk(t)|p�dt → 0 for k → ∞.

In view of (7.29), the assertion is proved. �

The following modification of Aubin-Lions’ Lemma 7.7 is useful in the situa-tion that we have another Banach space; let us denote it byH , and an L∞-estimatevalued in H at our disposal. This enables us to improve integrability in the targetspace from p to p/λ:

Lemma 7.8 (Interpolation). Let V1, V2, V3 be as in Lemma 7.7, and H andV4 other Banach spaces such that V1 � V2 ⊂ V4 ⊂ H and (V2, V4, H) forms aninterpolation triple in the sense∥∥v∥∥

V4≤ C

∥∥v∥∥1−λ

H

∥∥v∥∥λV2

(7.36)

for some λ ∈ (0, 1); then

W 1,p,q(I;V1, V3) ∩ L∞(I;H) � Lp/λ(I;V4). (7.37)

Proof. By (7.36), we have the estimate∫ T

0

∥∥v(t)∥∥p/λV4

dt ≤ Cp/λ

∫ T

0

∥∥v(t)∥∥pV2

∥∥v(t)∥∥(1−λ)p/λ

Hdt

≤ Cp/λ∥∥v∥∥(1−λ)p/λ

L∞(I;H)

∥∥v∥∥pLp(I;V2)

(7.38)

with C the constant from (7.36). Then by (7.38) we get∥∥uk − u∥∥Lp/λ(I;V4)

≤ C∥∥uk − u

∥∥1−λ

L∞(I;H)

∥∥uk − u∥∥λLp(I;V2)

→ 0 (7.39)

because ‖uk−u‖1−λL∞(I;H) is bounded by assumption while ‖uk−u‖λLp(I;V2)

converges

to 0 by Lemma 7.7. �

Still one modification of Aubin-Lions’ Lemma 7.7 will be found useful.


Corollary 7.9 (Generalization for dudt a measure13). Assuming V1 � V2 ⊂ V3

(the compact and the continuous embeddings between Banach spaces, respectively),V1 reflexive, the Banach space V3 having a predual space V ′3 , i.e. V3 = (V ′3)

∗, and1 < p < +∞, it holds that

W 1,p,M (I;V1, V3) :={u∈Lp(I;V1);

du

dt∈M (I;V3)

}� Lp(I;V2). (7.40)

Proof. By Hahn-Banach’s Theorem 1.5, we can extend ddtuk from M (I;V3) =

C(I;V ′3)∗ to L∞(I;V ′3)

∗ while preserving its norm, so that we can test ddtuk by the

discontinuous function ψδ as done in (7.31), and eventually get the same estimateas (7.35) but with

∥∥ ddtuk

∥∥L∞(I/2+δ;V ′

3 )∗ instead of

∥∥ ‖ ddtuk‖V3

∥∥L1(I/2+δ)

. �

The case p = +∞ has been excluded in Lemma 7.7 as well as in Corollary 7.9.There is, however, a simple modification for weakly continuous functions.

Lemma 7.10 (Arzela-Ascoli-type modification). Let V1, V2 and V3 be as inLemma 7.7 with V3 a separable reflexive Banach space and let 1 < q ≤ +∞. Then

C(I; (V1,weak)) ∩W 1,q(I;V3) � C(I;V2). (7.41)

Proof. Consider a bounded sequence {uk}k∈N in C(I; (V1,weak))∩W 1,q(I;V3). Itsuffices to make the proof for q < +∞. As W 1,q(I;V3) is reflexive, we can takea subsequence (denoted again as {uk}k∈N for simplicity) such that uk ⇀ u inW 1,q(I;V3). By Lemma 7.1, W 1,q(I;V3) ⊂ C(I;V3), hence uk(t) ⇀ u(t) in V3 forany t ∈ I. Then also uk(t) ⇀ u(t) in V1, and by the compact embedding V1 � V2

also uk(t) → u(t) strongly in V2 for any t ∈ I. The sequence {uk : I → V3}k∈N isequicontinuous because

∥∥uk(t1)−uk(t2)∥∥V3

≤∥∥∥ ∫ t2

t1

duk

dtdt∥∥∥V3

≤∫ t2

t1

∥∥∥duk

dt

∥∥∥V3

dt

≤∥∥∥duk

dt

∥∥∥Lq(I;V3)

‖1‖Lq′([t1,t2]) = |t1−t2|1−1/q∥∥∥duk

dt

∥∥∥Lq(I;V3)

for any 0 ≤ t1 < t2 ≤ T . Assume that a selected sequence {uk}k∈N does notconverge to u in C(I;V2). Thus ‖uk−u‖C(I;V2) ≥ ε > 0 for some ε and for all k(from the already selected subsequence), and we would get ‖uk(tk)−u(tk)‖V2 ≥ εfor some tk. By compactness of I := [0, T ], we can further select a subsequenceand some t ∈ I so that tk → t. Then we have u(tk) → u(t) in V2. By theabove proved equicontinuity, we have also uk(tk) ⇀ u(t) in V3. By the bound-edness of {uk(tk)}k∈N in V1 � V2, we have also uk(tk) → u(t) in V2. Then‖uk(tk)−u(tk)‖V2 → ‖u(t)−u(t)‖V2 = 0, a contradiction. �

13For p = +∞ in Lp(I; V1), this assertion has been stated in [310].

Chapter 8

Evolution by pseudomonotoneor weakly continuous mappings

As already advertised in the previous Chapter 7, evolution problems involve onevariable, a time t, having a certain specific character and thus a specific treatmentis useful, although some methods (applicable under special circumstances, seeSections 8.9 and 8.10) can wipe this specific character off. Conventional methodswe will scrutinize in this chapter deal with this one-dimensional variable t by twoways:(i) discretize t, and then thus created auxiliary approximate problems are based

on our knowledge from Part I,

(ii) keep t continuous but approximate the rest by a Galerkin method similarlyas we did in Section 2.1, and then the approximate problems are based onordinary differential equations and Section 1.6.

8.1 Abstract initial-value problems

In this chapter, we will focus on the setting that the evolution is governed by theabstract initial-value problem (the so-called abstract Cauchy problem):1

du

dt+A

(t, u(t)

)= f(t) for a.a. t∈I, u(0) = u0 . (8.1)

The latter equality in (8.1) is called an initial condition.. We will address especiallythe case that A : I × V → V ∗, I := [0, T ] a fixed bounded time interval, andV ⊂ H ∼= H∗ ⊂ V ∗ is a Gelfand triple, V a separable reflexive Banach space

1In fact, making A time dependent allows us to consider f = 0 without loss of generality.Anyhow, it is often convenient to distinguish f . Also, the adjective “Cauchy” in concrete par-tial differential equations often refers to initial-value problems on unbounded domains withoutboundary conditions.


213

214 Chapter 8. Evolution by pseudomonotone or weakly continuous mappings

embedded continuously and densely (and, for treatment of non-monotone lower-order terms, also compactly) into a Hilbert space H ∼= H∗.

Definition 8.1 (Strong solution). We call u ∈ W 1,p,p′(I;V, V ∗) with p′ := p/(p− 1)

a strong solution to (8.1) if the first equality in (8.1) holds in V ∗ while the secondone in H .

The main feature of the concept of Definition 8.1 is that ddtu lives in the dual

space to the space where u lives, i.e. here Lp(I;V ). In particular, the initial con-dition u(0) = u0 ∈ H has a good sense simply due to Lemma 7.3. Sometimes, thisinformation about d

dtu is not available, however. Then, in some cases, it suffices totake Lp(I;V ) in Definition 8.1 smaller, e.g. Lp(I;V )∩Lq(I;H) as in Remark 8.12or W 1,p,p′

(I;V, V ∗)∩L∞(I;H), but sometimes even this is not possible. Such situ-ations are indeed difficult but some results can still be achieved with the followingdefinition, working with a dense subspace Z ⊂ V and considering A : I×V → Z∗.

Definition 8.2 (Weak solution). We call u ∈ Lp(I;V ) ∩ C(I; (H,weak)) a weaksolution to (8.1) if the integral identity∫ T

0

⟨A(t, u(t)

)−f(t), v(t)⟩Z∗×Z

−⟨dvdt

, u(t)⟩V ∗×V

dt+(v(T ), u(T )

)=(v(0), u0

)(8.2)

holds for all v∈W 1,∞,∞(I;Z, V ∗); the parenthesis (·, ·) is the inner product in H .

Sometimes, one can consider a modification of Definition 8.2 by requiringv(T ) = 0 and then u∈Lp(I;V ) only. A justification of Definition 8.2 is its selec-tivity (Lemma 8.4 with a uniqueness in qualified cases in Theorem 8.36 below)and the following assertion of consistency:

Lemma 8.3 (Consistency of the weak solution). Any strong solution u to(8.1) with f ∈ Lp′

(I;V ∗) is also a weak solution (considering an arbitrary denseZ ⊂ V ).

Proof. Note that t �→ A(t, u(t)) = f(t) − ddtu ∈ Lp′

(I;V ∗). Considering v ∈W 1,∞,∞(I;Z, V ∗) ⊂ W 1,p,p′

(I;V, V ∗) and testing (8.1) by v = v(t), we obtain(8.2) after integration over I by using the by-part formula (7.15) and the initialcondition u(0) = u0. �

Lemma 8.4 (Selectivity of the weak solution). Let f(t) ∈ V ∗ for a.a. t ∈ I.Then any weak solution u which also belongs to W 1,p,p′

(I;V, V ∗) and for whichA(t, u(t)) ∈ V ∗ for a.a. t ∈ I is also the strong solution due to Definition 8.1.

Proof. The qualification of u allows us to use the by-part formula (7.15) to theterm 〈 d

dtv, u〉 in (8.2), which results in∫ T

0

⟨dudt

+A(t, u(t))− f(t), v(t)⟩V ∗×V

dx =(u0 − u(0), v(0)

). (8.3)

8.2. Rothe method 215

Using v(0) = 0, the right-hand-side term vanishes, and realizing that theleft-hand-side integral is the duality pairing in Lp′

(I;V ∗) × Lp(I;V ) we ob-tain d

dtu + A(t, u(t)) = f(t) for a.a. t ∈ I, cf. Exercise 8.49 and realize that

{g ∈ W 1,p′(I); g(0) = 0} is dense in Lp(I). Putting this into (8.3), we obtain

(u0 − u(0), v(0)) = 0. Taking now v general, we get still u(0) = u0. �

It should be emphasized that various adjectives such as “weak” or “strong”may address different issues, depending on the context. This is a general state ofthe art in theory of partial differential equations, especially evolutionary, which isreflected also here. Therefore, referring to a specific definition is always advisable.For readers convenience, the used terminology is summarized in Table 2.

abstract level of concrete differential equationslevel or variational inequalities

strong solution weak solutionweak solution very weak solution

Table 2. Terminology about solutions to evolution problems.

8.2 Rothe method

Let us begin with the problem (8.1) in the autonomous case, i.e. A : V → V ∗ isindependent of time t:

du

dt+A

(u(t)

)= f(t) , u(0) = u0 . (8.4)

In this section we present the so-called Rothe method [356] consisting in discretiza-tion in time. Let τ > 0 be a time step; for simplicity, we take an equidistant parti-tion of I and suppose T/τ is an integer. Moreover, we will work with a sequence ofsuch time steps {τl}l∈N such that liml→∞ τl = 0 and, again for simplicity, assumethat τl = 2−lT so that the partitions are “nested” in the sense that the subsequentpartition always refines the previous one. Let us further agree to omit the indexl and write simply τ → 0 instead of τl → 0 for l → ∞. Moreover, we must ap-proximate values of f at particular points t = kτ , 0 ≤ k ≤ T/τ . One possible wayis to apply a mollifier as (7.10) to f instead of u; let us denote by fε ∈ C(I;V ∗)the resulting smoothened right-hand side. Then, choosing a suitable ε = ε(τ), cf.Lemma 8.7 below, we put fk

τ = fε(τ)(kτ). Then we define ukτ ∈ V , k = 1, . . . , T/τ ,

by the following recursive formula:

ukτ − uk−1

τ

τ+A(uk

τ ) = fkτ , fk

τ := fε(τ)(kτ) , u0τ = u0τ , (8.5)

called sometimes an implicit Euler formula.2 Sometimes, the recursion (8.5) isstarted simply from the original initial condition for (8.4), i.e. u0τ = u0, but in

2The adjective “implicit” emphasizes that the unknown ukτ occurs in A and cannot be explic-

itly expressed, and to distinguish it from an explicit Euler formula ukτ −uk−1

τ + τA(uk−1τ ) = τfk

τ


general u0τ may be only a suitable approximation of u0, cf. (8.38) below. Further-more, we define the piecewise affine interpolant uτ ∈ C(I;V ) by

uτ (t) =( t

τ− (k−1)

)ukτ +

(k − t

τ

)uk−1τ for (k−1)τ < t ≤ kτ (8.6)

and the piecewise constant interpolant uτ ∈ L∞(I;V ) by

uτ (t) = ukτ for (k−1)τ < t ≤ kτ, k = 0, . . . , T/τ. (8.7)

Let us note that uτ has a derivative ddtuτ ∈ L∞(I;V ) which is piecewise constant;

cf. Figure 17, whereas uτ has only a distributional derivative composed from Diracmasses, cf. (8.34). Analogously, fτ abbreviates the piece-wise constant functiondefined by

fτ (t) := fkτ := fε(τ)(kτ) for t ∈ [(k−1)τ, kτ

], k = 1, . . . , T/τ. (8.8)

calculated values piecewise affinepiecewise constantinterpolant uτinterpolant uτ

time derivative duτdt

u0τ

u1τ

u2τ

VVV V

TT TT 00 00τ

Figure 17. Illustration of Rothe’s interpolants uτ and uτ constructed from a sequence{uk

τ}T/τk=0, and the time derivative d

dtuτ ; the dashed line on the last picture

shows the interpolated derivative [ ddtuτ ]

i which will be used in Chapter 11.

Let us consider a seminorm on V , denoted by |·|V , such that the following “abstractPoincare-type” inequality holds:

∃CP∈ R+ ∀v∈V : ‖v‖V ≤ C

P

(|v|V + ‖v‖H). (8.9)

A trivial case of (8.9) is | · |V := ‖ · ‖V with CP= 1. Referring to such seminorm,

we will call A semi-coercive3⟨A(u), u

⟩ ≥ c0|u|pV − c1|u|V − c2‖u‖2H ; (8.10)

this condition essentially determines the power p < +∞ in the functional settingof the problem. In some special cases, typically for | · |V = ‖ · ‖V and c1 = 0, themappings A satisfying (8.10) are also called weakly coercive, cf. e.g. [314], and,in the case p = 2, it also generalizes the so-called Garding inequality (designedoriginally for specific linear differential operators [172]). In this Chapter, we willassume p > 1.

which is, however, not suitable if V is infinite-dimensional. For semi-implicit formulae see Re-mark 8.14 below, while a multilevel formula is in Remark 8.20.

3Note that, even if |·|V = ‖·‖V would be considered, 〈A(u), u〉may tend to −∞ for ‖u‖V →∞provided p < 2. Thus (8.10) is indeed much weaker than the “full” coercivity (2.5). For someconsiderations, (8.10) can be even weakened by considering c2‖u‖2H ln(‖u‖2H ) instead of c2‖u‖2Hand then using a generalized Gronwall inequality, though e.g. Lemma 8.5 would not hold.


Lemma 8.5 (Existence of Rothe’s sequence). Let A : V → V ∗ be pseudo-monotone and semi-coercive, f ∈L1(I;V ∗), and u0τ ∈V ∗. Then, for a sufficientlysmall τ > 0 (namely τ < 1/c2), the Rothe solution uτ ∈C(I;V ∗) does exist.

Proof. Let us notice that the identity mapping I : V → V is monotone as amapping V → V ∗ because the embedding V ⊂ H ⊂ V ∗ implies⟨

Iu− Iv, u− v⟩V ∗×V

=⟨u− v, u− v

⟩H∗×H

= (u− v, u− v) = ‖u− v‖2H ≥ 0.

Thus 1τ I + A : V → V ∗ is pseudomonotone because I is monotone, bounded,

radially continuous (hence pseudomonotone by Lemma 2.9) and the sum of twopseudomonotone mappings is again pseudomonotone, see Lemma 2.11(i). Also,1τ I+A : V → V ∗ is coercive for τ small enough. Indeed,⟨

u+ τA(u), u⟩= (u, u) + τ

⟨A(u), u

⟩ ≥ τc0|u|pV − τc1|u|V + (1− τc2)‖u‖2Hwhich can, for τc2 ≤ 1, be estimated from below by ε(|u|V +‖u‖H)min(2,p)−1/ε ≥ε(‖u‖V /CP

)min(2,p) − 1/ε for ε > 0 small enough, so that the coercivity (2.5) isfulfilled. Thus, by Theorem 2.6, the equation [ 1τ I+A](u) = 1

τ uk−1τ + fk

τ has somesolution u =: uk

τ ∈ V . �

The mapping A : V → V ∗ induces a superposition mapping (i.e. a specialcase of the abstract Nemytskiı mapping) A by[

A (u)](t) := A

(u(t)

). (8.11)

In the following, we will need to have some information about the time derivativewhich can be ensured by assuming that

A : Lp(I;V ) ∩ L∞(I;H) → Lq′(I;Z∗) bounded (8.12)

with some q ≥ 1 and Z ⊂ V densely. A simple example for a condition thatguarantees (8.12) is the growth condition on A:

∃C : R → R increasing ∀v∈V :∥∥A(v)∥∥

Z∗ ≤ C(‖v‖H)(1 + ‖v‖p/q′V

). (8.13)

In concrete cases, (8.12) may involve rather fine estimates, cf. Example 8.63(2)below.

Lemma 8.6 (Basic a-priori estimates). Let (8.9) hold, A be pseudomonotoneand semi-coercive, let

f = f1 + f2 with f1∈Lp′(I;V ∗), f2∈Lq′(I;H), (8.14)

let the mollified approximation fε(τ) = (f1+ f2)ε(τ) := f1ε(τ)+ f2ε(τ) used in (8.5)and the corresponding interpolant fτ = f1τ + f2τ constructed by (8.5) satisfy∥∥f1ε(τ)∥∥C(I;V ∗) ≤

K1√τ

and∥∥f2ε(τ)∥∥C(I;H)

≤ K2√τ, (8.15a)∥∥f1τ∥∥Lp′(I;V ∗) ≤ K1 and

∥∥f1τ∥∥Lq′ (I;H)≤ K2 (8.15b)


for some (but arbitrarily chosen) K1, K2 and let {u0τ}τ>0 be bounded in H. Then,for any 0 < τ ≤ τ0 with τ0 small enough so that

2τ0c2 +√τ0CP

K1 +√τ0K2 < 1 (8.16)

with CPfrom (8.9), the following a-priori estimates

‖uτ‖C(I;H) ≤ C1 , (8.17a)

‖uτ‖L∞(I;H) ∩ Lp(I;V ) ≤ C1 , (8.17b)∥∥uτ |[τ0,T ]

∥∥Lp([τ0,T ];V )

≤ C1 , (8.17c)∥∥∥duτ

dt

∥∥∥L2(I;H)

≤ C2√τ

, (8.17d)

hold with some C1 and C2 depending on p, CP, ‖f1‖Lp′(I;V ∗), ‖f2‖L1(I;H), and

supτ>0 ‖u0τ‖H only. Moreover, if u0τ ∈ V , then

‖uτ‖Lp(I;V ) ≤ p

√Cp

1 + τ‖u0τ‖pV . (8.18)

Eventually, if also (8.12) and (8.14) hold with q ≥ p, then∥∥∥duτ

dt

∥∥∥Lq′ (I;Z∗)

≤ C3, (8.19a)∥∥∥duτ

dt

∥∥∥M (I;Z∗)

≤ T 1/qC3. (8.19b)

Before starting a rigorous proof, let us sketch the heuristics in an easilyobservable way neglecting (otherwise necessary) technicalities related to the ap-proximate problem. First, “multiply” the equation d

dtu+A(u) = f by the solution

u itself, and then use 〈 ddtu, u〉 = 1

2ddt‖u‖2H , cf. also (7.23), the semi-coercivity (8.10)

and Young’s inequality. This gives

d

dt

(12‖u‖2H + c0

∫ t

0

|u(θ)|pV dθ)=

1

2

d

dt‖u‖2H + c0|u|pV ≤

⟨dudt

+A(u), u⟩

+ c1|u|V + c2‖u‖2H = c1|u|V + c2‖u‖2H +⟨f, u

⟩=: R1 +R2 +R3, (8.20)

where |u|V , ‖u‖H , f , etc. abbreviate |u(t)|V , ‖u(t)‖H , f(t), etc., respectively. By(8.9), the last term can be estimated as

R3 :=⟨f, u

⟩=⟨f1 + f2, u

⟩ ≤ ∥∥f1∥∥V ∗∥∥u∥∥

V+∥∥f2∥∥H∥∥u∥∥H

≤ CP

∥∥f1∥∥V ∗(∣∣u∣∣

V+∥∥u∥∥

H

)+∥∥f2∥∥H(12 +

1

2

∥∥u∥∥2H

)≤ Cp′

PCε

∥∥f1∥∥p′

V ∗ + CpPε∣∣u∣∣p

V+(C

P

∥∥f1∥∥V ∗ +∥∥f2∥∥H)(12 +

1

2

∥∥u∥∥2H

)(8.21)


with Cε from (1.22). The term CPε|u|pV can then be absorbed in the left-handside if ε < 1

2c0/CPis chosen, while the other terms have integrable coefficients as

functions of t just by the assumption (8.14). Similarly, R1 ≤ c1(Cε + ε|u|pV ) withCε again from (1.22), and then c1ε|u|pV can be absorbed in the left-hand side ifalso ε < c0/(2c1). Altogether, we obtain

d

dt

(1

2‖u‖2H +

(c0 − εC

P− εc1

)∫ t

0

∣∣u(θ)∣∣pVdθ

)≤ Cε

(c1 + C

P‖f1‖p

′V ∗)

+c2‖u‖2H +(CP

∥∥f1∥∥V ∗ +∥∥f2∥∥H)(12 +

1

2

∥∥u∥∥2H

). (8.22)

Then we can use directly Gronwall’s inequality (1.66). In such a way, we obtain

‖u(t)‖2H +∫ t

0 |u(θ)|pV dθ bounded uniformly with respect to t ∈ I, which yields

already (8.17a). Then, for t = T , we get also the bound for∫ T

0 |u(t)|pV dt, so thatstill (8.17b) follows by using also (8.10) and the already obtained estimate (8.17a).The “dual” estimate (8.19a) is then essentially determined by (8.17b) through thecondition (8.12). Assuming we know d

dtu = f − A(u), which we indeed will knowfor the discrete problem, and using Holder’s inequality, we have

∥∥∥dudt

∥∥∥Lq′ (I;Z∗)

= sup‖v‖Lq(I;Z)≤1

∫ T

0

⟨dudt

, v⟩dt


∫ T

0

⟨f −A(u), v

⟩dt

≤ sup‖v‖Lq(I;Z)≤1

(‖f‖Lq′(I;Z∗)+ ‖A(u)‖Lq′(I;Z∗)

)‖v‖Lq(I;Z) (8.23)

and then (8.12) with the already proved estimate of u in Lp(I;V ) ∩ L∞(I;H) isused.

On the other hand, the estimates (8.17c,d) and (8.18) explicitly involve τand τ0 and cannot be seen by such heuristical considerations.

Proof of Lemma 8.6. Let us now make the proof of Lemma 8.6 with full rigor.Multiply (8.5) by uk

τ . This yields⟨ukτ − uk−1

τ

τ, uk

τ

⟩+⟨A(uk

τ ), ukτ

⟩=⟨fkτ , u

kτ

⟩. (8.24)

Then sum (8.24) for k = 1, . . . , l, multiply by τ , and use the identity (ukτ −

uk−1τ , uk

τ ) = ‖ukτ‖2H − (uk−1

τ , ukτ ) =

12‖uk

τ‖2H − 12‖uk−1

τ ‖2H + 12‖uk

τ − uk−1τ ‖2H which

follows from (1.4)4 and which implies the estimate

4Indeed, using u := uk−1τ and v := uk

τ in (1.4) yields (uk−1τ , uk

τ ) =14‖uk

τ +uk−1τ ‖2H − 1

4‖uk

τ −uk−1τ ‖2H = 1

4‖uk

τ‖2H + 12(uk−1

τ , ukτ ) +

14‖uk−1

τ ‖2H − 14‖uk

τ − uk−1τ ‖2H which further yields the

identity in question.


l∑k=1

⟨ukτ − uk−1

τ , ukτ

⟩=

l∑k=1

(ukτ − uk−1

τ , ukτ

)=

l∑k=1

1

2‖uk

τ‖2H − 1

2‖uk−1

τ ‖2H

+1

2‖uk

τ − uk−1τ ‖2H ≥ 1

2‖ul

τ‖2H − 1

2‖u0

τ‖2H . (8.25)

This gives

1

2

∥∥ulτ

∥∥2H− 1

2

∥∥u0τ

∥∥2H+ τ

l∑k=1

⟨A(uk

τ ), ukτ

⟩ ≤ τl∑

k=1

⟨fkτ , u

kτ

⟩. (8.26)

Following the scheme (8.20)–(8.22), by (8.10) and by Holder’s inequality, we canfurther estimate

1

2‖ul

τ‖2H + c0τ

l∑k=1

|ukτ |pV ≤ 1

2‖u0τ‖2H + τ

l∑k=1

(〈fk

τ , ukτ 〉+c1|uk

τ |V +c2‖ukτ‖2H

)

≤ 1

2‖u0τ‖2H + τ

l∑k=1

(ε(c1 + C

P

)|ukτ |pV + Cε

(C1 + C

P

∥∥fk1τ

∥∥p′

V ∗)

+1

2CP

∥∥fk1τ

∥∥V ∗+

1

2

∥∥fk2τ

∥∥H+(c2+

1

2CP

∥∥fk1τ

∥∥V ∗+

1

2

∥∥fk2τ

∥∥H

)∥∥ukτ

∥∥2H

)(8.27)

with ε > 0 small and Cε correspondingly large, cf. (1.22). If ε < c0/(c1 + CP), we

can absorb the term with ε(c1 + CP)|ukτ |pV in the respective term in the left-hand

side. Then discrete Gronwall’s inequality (1.70) can be used; note that (1.70) hererequires

τ <1

2c2 +maxk=1,...,T/τ

(C

P

∥∥fk1τ

∥∥V ∗ +

∥∥fk2τ

∥∥H

) (8.28)

which is indeed satisfied if τ ≤ τ0 with τ0 small as specified in (8.16). Also notethat

τ

l∑k=1

∥∥fk1τ

∥∥p′

V ∗ ≤ τ

T/τ∑k=1

∥∥fk1τ

∥∥p′

V ∗=∥∥f1τ∥∥p′

Lp′(I;V ∗) ≤ Kp′1 , (8.29a)

τ

l∑k=1

∥∥fk2τ

∥∥H

≤ τ

T/τ∑k=1

∥∥fk2τ

∥∥H=∥∥f2τ∥∥L1(I;H)

≤ T 1/qK2 (8.29b)

for K1 and K2 from (8.15b). By this way, we get already the estimate of‖uτ‖L∞(I;H)∩Lp(I;V ) and of ‖uτ‖L∞(I;H) if τ ≤ τ0. Certainly

5

‖uτ‖Lp([τ,T ];V ) ≤ ‖uτ‖Lp(I;V ), (8.30)

5It can easily be proved for p = 1 and for p = +∞. For 1 < p < +∞, we get it by interpolation.


which already gives (8.17c) for τ ≤ τ0.Moreover, by a finer usage of (8.25) exploiting also the “forgotten” term

‖ukτ − uk−1

τ ‖2H , we get still the boundedness of∑l

k=1 ‖ukτ − uk−1

τ ‖2H , from which(8.17d) follows.

As to (8.18), we can formally extend uτ for t ≤ 0 by u0τ , and then, like(8.30), we have

‖uτ‖Lp(I;V ) ≤ ‖uτ‖Lp([−τ,T ];V ) =(∫ 0

−τ

‖u0τ‖pV dt+

∫ T

0

‖uτ‖pV dt)1/p

=(τ‖u0τ‖pV + ‖uτ‖pLp(I;V )

)1/p. (8.31)

Then (8.17b) gives (8.18).As to (8.19a), in view of (8.5), as in (8.23), we have∫ T

0

⟨duτ

dt, v⟩dt =

∫ T

0

⟨fτ −A(uτ ), v

⟩dt

≤(‖fτ‖Lq′ (I;Z∗) + ‖A(uτ )‖Lq′ (I;Z∗)

)‖v‖Lq(I;Z)

≤(‖f‖Lq′(I;Z∗) +

(∫ T

0

∥∥A(uτ )∥∥q′Z∗ dt

)1/q′)‖v‖Lq(I;Z) (8.32)

where fτ is from (8.8). As Lq′(I;Z∗) is isometrically isomorphic with Lq(I;Z)∗,cf. Proposition 1.38, it holds that∥∥∥duτ

dt

∥∥∥Lq′ (I;Z∗)


∫ T

0

⟨duτ

dt, v⟩dt (8.33)

≤ N‖f‖Lp′(I;V ∗) + sup‖v‖Lp(I;V )∩L∞(I;H)≤C1

∥∥A (v)∥∥Lq′ (I;Z∗)

where N denotes the norm of the embedding Lp′(I;V ∗) ⊂ Lq′(I;Z∗); here the

assumption q ≥ p is used. The a-priori bound (8.17b) then gives (8.19a). Then(8.19b) follows easily:

∥∥∥duτ

dt

∥∥∥M (I;Z∗)

=∥∥∥ T/τ∑

k=1

(ukτ−uk−1

τ )δ(k−1)τ

∥∥∥M (I;Z∗)

= τ

T/τ∑k=1

∥∥∥ukτ−uk−1

τ

τ

∥∥∥Z∗

=∥∥∥duτ

dt

∥∥∥L1(I;Z∗)

≤ T 1/q∥∥∥duτ

dt

∥∥∥Lq′ (I;Z∗)

≤ T 1/qC3 (8.34)

where δ(k−1)τ denotes the Dirac distribution at time t = (k−1)τ . �

For the convergence, the following approximation property will be founduseful:


Lemma 8.7 (Convergence of fτ). If f ∈ Lq(I;X) for 1 ≤ q < +∞ and Xa Banach space, then fτ defined by (8.8) with fε being a mollified f satisfying‖fε‖C1(I;X) ≤ L(ε) for some L : R+ → R+ (possibly unbounded) and with ε = ε(τ)so that both limτ→0 ε(τ) = 0 and supτ>0

√τL(ε(τ)) < +∞ converges to f in

Lq(I;X) when τ → 0. Also ‖fτ‖C(I;X) ≤ Kτ−1/2, cf. also (8.16), holds for someK, too.

Proof. As fε : I → X is Lipschitz continuous with the Lipschitz constant L(ε), itholds that ‖fε−fτ‖C(I;X) ≤ L(ε)τ . Choosing ε = ε(τ) so that τL(ε(τ)) → 0, wehave

‖fτ−f‖Lq(I;X) ≤ ‖fτ−fε(τ)‖Lq(I;X) + ‖fε(τ)−f‖Lq(I;X)

≤ T 1/q‖fτ−fε(τ)‖L∞(I;X) + ‖fε(τ)−f‖Lq(I;X)

≤ T 1/qτL(ε(τ)) + ‖fε(τ)−f‖Lq(I;X) → 0

because fε → f in Lq(I;X), cf. the proof of Lemma 7.2. Eventually, ‖fτ‖C(I;X) ≤L(ε(τ)) ≤ Kτ−1/2 is obvious with K := supτ>0

√τL(ε(τ)). �

When one uses Lemma 8.7 for f1 ∈ Lp′(I;V ∗) and also for f2 ∈ Lq′(I;H) in

place of f ∈ Lq(I;X), it will justify (8.15a) for ε = ε(τ) decreasing to 0 sufficientlyslowly for τ → 0 while also (8.15b) holds because of the proved convergence. Anadditional useful auxiliary assertion addresses pseudomonotonicity of A .

Lemma 8.8 (Papageorgiou [323], here modified6). Let A : V → V ∗ be pseu-domonotone in the sense of (2.3), satisfying (8.10) and (8.13) with q = p andZ = V . Then A is pseudomonotone on W := W 1,p,M (I;V, V ∗)∩L∞(I;H) in thesense analogous to (2.3), i.e.,

A is bounded, and (8.35a)

uk∗⇀ u in W

lim supk→∞

⟨A (uk), uk−u

⟩ ≤ 0

}⇒

∀v∈W :⟨A (u), u−v

⟩ ≤ lim infk→∞

⟨A (uk), uk−v

⟩.(8.35b)

Proof. The boundedness (8.35a) of A : W → W ∗ follows just by (8.4).To prove (8.35b), let us now take uk

∗⇀ u in W . By Helly’s selection principlefor mappings valued in a separable reflexive V ∗ (see e.g. [39, Chap.1, Thm.3.5]),there is a subsequence (denoted by the same indexes k for simplicity) and u :I → V ∗ with a bounded variation such that uk(t) ⇀ u(t) in V ∗ for all t ∈ I.Yet, u = u a.e. on I; indeed, for any v ∈ L∞(I;V ∗) we have 〈uk, v〉 → 〈u, v〉and, by Lebesgue Theorem 1.14, simultaneously 〈uk, v〉 → 〈u, v〉. Thus, using theboundedness of {uk(t)}k∈N in H for a.a. t∈I, we have even uk(t) ⇀ u(t) in H fora.a. t ∈ I.

6In [323], a non-autonomous case like in Lemma 8.29 below has been addressed but in the

case W := W 1,p,p′(I;V, V ∗). See also [209, Part II, Chap.I, Thm.2.35].


Denote ξk(t) := 〈A(uk(t)), uk(t)− u(t)〉 and, in accord with (8.35b), assume

lim supk→∞∫ T

0 ξk(t) dt ≤ 0. By (8.10) and (8.13) with q = p and Z = V , we have

ξk(t) ≥ c0∣∣uk(t)

∣∣pV− ζk(t), with

ζk(t) := c1∣∣uk(t)

∣∣V+ c2

∥∥uk(t)∥∥2H+ C

(‖uk(t)‖H)(1+‖uk(t)‖p−1

V

)∥∥u(t)∥∥V. (8.36)

Now, for a moment, assume lim infk→∞ ξk(t) < 0 with t fixed. Then, for a subse-quence such that limk→∞ ξk(t) < 0, the estimate (8.36) implies that {uk(t)}k∈Nis bounded in V , hence (again for a subsequence depending possibly on t)uk(t) ⇀ u(t) in V because also uk(t) ⇀ u(t) in H . By pseudomonotonicity (2.3)of A used with uk = uk(t) and u = v = u(t), we obtain lim infk→∞ ξk(t) ≥ 0. Thisholds for a.a. t ∈ I.

As ξk ≥ −ζk and {ζk}k∈N is uniformly integrable7, by the generalized FatouTheorem 1.18 it holds

0 ≤∫ T

0

lim infk→∞

ξk(t) dt ≤ lim infk→∞

∫ T

0

ξk(t) dt ≤ lim supk→∞

∫ T

0

ξk(t) dt ≤ 0, (8.37)

and therefore limk→∞∫ T

0 ξk(t) dt = 0.

Since lim infk→∞ ξk(t) ≥ 0, we have ξ−k (t) → 0 a.e. and thus, by Vitali’s

Theorem 1.17, we also have limk→∞∫ T

0ξ−k (t) dt = 0 because 0 ≥ ξ−k ≥ −ζk

and because {ζk}k∈N is uniformly integrable. Altogether, limk→∞∫ T

0|ξk(t)|dt =

limk→∞∫ T

0ξk(t) − 2ξ−k (t) dt = 0. Hence, possibly in terms of a subsequence,

limk→∞ ξk(t)= 0 for a.a. t ∈ I. Taking v ∈ W , by the pseudomonotonicity of A,we have lim infk→∞〈A(uk(t)), uk(t) − v(t)〉 ≥ 〈A(u(t)), u(t) − v(t)〉 for a.a. t ∈ I,and eventually again by the generalized Fatou Theorem 1.18 it holds

lim infk→∞

⟨A (uk), uk − v

⟩= lim inf

k→∞

∫ T

0

⟨A(uk(t)), uk(t)− v(t)

⟩dt

≥∫ T

0

lim infk→∞

⟨A(uk(t)), uk(t)−v(t)

⟩dt ≥

∫ T

0

⟨A(u(t)), u(t)−v(t)

⟩dt =

⟨A (u), u−v

⟩because 〈A(uk(·), uk(·)−v(·))〉 has a uniformly integrable minorant, namely −ζkas in (8.36) but with v in place of u. �

As we required ddtu ∈ Lp′

(I;V ∗) in Definition 8.1, it is reasonable to have

both A (u) and f in Lp′(I;V ∗), i.e. q = p in (8.12) and (8.14). As always H ⊂ V ∗,

we can consider f2 = 0 in (8.14) without loss of generality as far as values of f

7The uniform integrability or, through Dunford-Pettis’ Theorem 1.16, rather equi-absolute-continuity of the collection {‖uk(·)‖p−1

V ‖u(·)‖V }k∈N can easily be seen by Holder inequality; in-

deed,∫J ‖uk(t)‖p−1

V ‖u(t)‖V dt ≤ (∫J ‖uk(t)‖pV dt)1/p

′(∫J ‖u(t)‖pV dt)1/p ≤ (

∫ T0 ‖uk(t)‖pV dt)1/p

′

(∫J‖u(t)‖pV dt)1/p can be made small uniformly if meas1(J) is small because of absolute-

continuity of ‖u(·)‖pV ∈ L1(I) and of boundedness of the collection {‖uk(·)‖pV }k∈N in L1(I).


concerns. For a more general integrability of f in time considering f2 �= 0 withq > p in (8.14), cf. Remark 8.12 below.

Theorem 8.9 (Existence of a strong solution). Let A : V → V ∗ be pseu-domonotone and semi-coercive, satisfy the growth condition (8.12) with q = p andZ = V , f ∈ Lp′

(I;V ∗) and u0 ∈ H. Then the Cauchy problem (8.4) possessesa strong solution u ∈ W 1,p,p′

(I;V, V ∗) which can, in addition, be attained in theweak topology of W 1,p,p′

(I;V, V ∗) by a subsequence of Rothe functions {uτ}τ>0

constructed by considering limτ→0 ε(τ) = 0 for fε(τ) in (8.5) satisfying (8.15a)(now with K2 = 0) and {u0τ}τ>0 ⊂ V such that

‖u0τ‖V = O(τ−1/p) and u0τ → u0 in H ; (8.38)

note that such {u0τ}τ>0 always exists because V is assumed dense in H.

Proof. Combining (8.18) with (8.38), we have uτ bounded in Lp(I;V ), andcounting also (8.17b) and (8.19a), we can take a subsequence and some u ∈L∞(I;H) ∩ Lp(I;V ), u ∈ L∞(I;H) ∩ Lp(I;V ), and u ∈ Lp′

(I;V ∗) such that

uτ∗⇀ u in L∞(I;H) ∩ Lp(I;V ) , (8.39a)

uτ∗⇀ u in L∞(I;H) ∩ Lp(I;V ) , (8.39b)

duτ

dt⇀ u in Lp′

(I;V ∗). (8.39c)

We want to show that

u = u &du

dt= u . (8.40)

Let us show that uτ − uτ ⇀ 0 in Lp(I;H). Take χ[τ0k1,τ0k2]v for some τ0 > 0,

k1 < k2 and v ∈ H ; linear combinations of all such functions are dense in Lp′(I;H)

due to Proposition 1.36. Then, for τ ≤ τ0,

⟨uτ − uτ , χ[τ0k1,τ0k2]v

⟩=

∫ τ0k2

τ0k1

⟨uτ (t)− uτ (t), v

⟩dt

=

k2τ0/τ∑k=k1τ0/τ+1

∫ kτ

(k−1)τ

⟨(uk

τ−uk−1τ )

t−kτ

τ, v⟩dt =

τ

2

k2τ0/τ∑k=k1τ0/τ+1

⟨ukτ−uk−1

τ , v⟩

=τ

2

⟨uk2τ0/ττ − uk1τ0/τ

τ , v⟩=

τ

2

⟨uτ (τ0k2)− uτ (τ0k1), v

⟩= O(τ),

where we eventually used that uτ (τ0k2) − uτ (τ0k1) is bounded in H by (8.17b).Thus uτ − uτ ⇀ 0 in Lp(I;H), and thus also in Lp(I;V ) because of (8.39). More-over, by using subsequently (8.39c), (7.15), and (8.39a), we get

〈u, ϕ〉 ←⟨duτ

dt, ϕ⟩= −

⟨uτ ,

dϕ

dt

⟩→ −

⟨u,

dϕ

dt

⟩(8.41)


for any ϕ ∈ D(I;V ), which, in particular, implies u = ddtu in the sense of distri-

butions8. Thus (8.40) must hold.The initial condition

u(0) = u0 (8.42)

is satisfied because uτ ⇀ u in W 1,p,p′(I;V, V ∗) and by the continuity (hence also

weak continuity) of u �→ u(0) : W 1,p,p′(I;V, V ∗) → H (see Lemma 7.3) we have

uτ (0) ⇀ u(0) in H so that

u0 ← u0τ = uτ (0) ⇀ u(0) , (8.43)

which immediately implies (8.42).From (8.5) one can see that d

dtuτ +A(uτ (t)) = fτ with fτ from (8.8). Thus,for any v ∈ Lp(I;V ), one has∫ T

0

⟨duτ

dt, v(t)

⟩+⟨A(uτ (t)), v(t)

⟩dt =

∫ T

0

⟨fτ (t), v(t)

⟩dt . (8.44)

In terms of 〈·, ·〉 as the duality between Lp′(I;V ∗) and Lp(I;V ), one can

rewrite (8.44) into 〈 ddtuτ , v〉 + 〈A (uτ ), v〉 = 〈fτ , v〉. Putting v − uτ instead of v,

one obtains⟨A (uτ ), v − uτ

⟩=⟨fτ , v − uτ

⟩− ⟨duτ

dt, v − uτ

⟩=: I(1)τ − I(2)τ . (8.45)

As fτ → f in Lp′(I;V ∗) due to Lemma 8.7 and uτ → u weakly in Lp(I;V ) due to

(8.39b) with (8.40), obviously

limτ→0

I(1)τ = 〈f, v − u〉. (8.46)

By (8.39a,c) with (8.40), the weak continuity of the mapping u �→ u(T ) :W 1,p,p′

(I;V, V ∗) → H (see Lemma 7.3), and by the weak lower semicontinuityof ‖ · ‖2H , one gets (using also (8.25))

lim supτ→0

I(2)τ ≤ lim supτ→0

(⟨duτ

dt, v⟩−⟨duτ

dt, uτ

⟩)≤ lim sup

τ→0

(⟨duτ

dt, v⟩− 1

2‖uτ (T )‖2H +

1

2‖u0τ‖2H

)= lim

τ→0

⟨duτ

dt, v⟩− 1

2lim infτ→0

‖uτ(T )‖2H +1

2limτ→0

‖u0τ‖2H

≤⟨dudt

, v⟩− ‖u(T )‖2H +

1

2‖u0‖2H =

⟨dudt

, v − u⟩

(8.47)

8Let us recall that ddt

u ∈ L (D(I), (V ∗,weak)) is defined by the Bochner integral [ ddt

u](φ) =

− ∫ T0 u( d

dtφ) dt for any φ ∈ D(I). The equality u = d

dtu can be got from (8.41) by putting

ϕ(t) = φ(t)v for φ ∈ D(I) and v ∈ V arbitrary.


where the last identity employs (8.42) and (7.22). Altogether, (8.46) and (8.47)lead to

lim infτ→0

⟨A (uτ ), v−uτ

⟩ ≥ ⟨f − du

dt, v − u

⟩. (8.48)

In particular, for v := u we have got lim supτ→0〈A (uτ ), uτ−u〉 ≤ 0. By Lemma 8.8,i.e. the pseudomonotonicity of A , we can conclude that, for any v ∈ Lp(I;V ),

lim infτ→0

⟨A (uτ ), uτ − v

⟩ ≥ ⟨A (u), u− v⟩. (8.49)

Joining (8.48) and (8.49), one gets 〈A (u), u− v〉 ≤ 〈f, u− v〉 − 〈 ddtu, u− v〉. As it

holds for v arbitrary, we can conclude that⟨A (u), v

⟩= 〈f, v〉 −

⟨dudt

, v⟩.

As v is arbitrary, A (u) = f − ddtu must hold for a.a. t ∈ I, cf. Exercise 8.49. �

Remark 8.10 (Error in uτ − uτ ). If (8.13) holds, then u = u in (8.40) can alterna-tively be proved by a simple direct calculation:

‖uτ − uτ‖q′

Lq′ (I;Z∗)=

T/τ∑k=1

∫ kτ

(k−1)τ

∥∥∥(ukτ − uk−1

τ )t−kτ

τ

∥∥∥q′Z∗

dt

=τ

q′+1

T/τ∑k=1

‖ukτ − uk−1

τ ‖q′Z∗ =τq

′+1

q′+1

T/τ∑k=1

∥∥∥ukτ − uk−1

τ

τ

∥∥∥q′Z∗

=τq

′

q′+1

∥∥∥duτ

dt

∥∥∥q′Lq′ (I;Z∗)

= O(τq′) (8.50)

where the bound (8.19a) has been used. Therefore, ‖uτ − uτ‖Lq′(I;Z∗) = O(τ)

and thus also ‖uτ − uτ‖L1(I;Z∗) = O(τ). As certainly Lp(I;V ) ⊂ L1(I;Z∗), wecan estimate the limit ‖u − u‖L1(I;Z∗) = 0 and thus the first equality in (8.40) isproved once again. Using (8.17d), the calculation (8.50) yields the error uτ − uτ

estimated in a stronger norm9:

‖uτ − uτ‖L2(I;H) =τ√3

∥∥∥duτ

dt

∥∥∥L2(I;H)

= O(√

τ). (8.51)

Remark 8.11 (Strong convergence uτ → u in Lp(I;V )). Let us subtract ddtuτ +

A(uτ (t)) = fτ from ddtu+A(u(t)) = f and test it by uτ−u. By using the inequality

〈 ddtuτ (t), uτ (t)− uτ (t)〉 = ‖ d

dtuτ‖2H(kτ−t) ≥ 0 for any t ∈ ((k−1)τ, kτ), we obtain⟨duτ

dt− du

dt, uτ − u

⟩=⟨duτ

dt, uτ−uτ

⟩+⟨duτ

dt− du

dt, uτ−u

⟩+⟨dudt

, uτ − uτ

⟩≥ 1

2

d

dt

∥∥uτ − u∥∥2H+⟨dudt

, uτ − uτ

⟩9See e.g. Feistauer [147, Theorem 8.7.25].


for a.a. t ∈ I; here the dualities are between V ∗ and V . After integration over I,this gives

1

2

∥∥uτ (T )− u(T )∥∥2H+⟨A (uτ )− A (u), uτ − u

⟩≤ 1

2‖u0τ − u0‖2H +

⟨fτ − f +

du

dt, uτ − uτ

⟩→ 0 (8.52)

by using respectively u0τ → u0 in H , fτ → f in Lp′(I;V ∗) due to Lemma 8.7, and

uτ − uτ ⇀ u − u = 0 in Lp(I;V ) due to (8.39a,b)–(8.40). This gives uτ → u inLp(I;V ) if A = A1+A2 and V is uniformly convex, and A1 is assumed d-monotonein the sense⟨

A1(u)− A1(v), u − v⟩Lp′(I;V ∗)×Lp(I;V )

≥(d(‖u‖Lp(I;V )

)− d(‖v‖Lp(I;V )

))(‖u‖Lp(I;V ) − ‖v‖Lp(I;V )

)(8.53)

for some d : R → R increasing, and A2 is totally continuous. Then we can useuniform convexity of Lp(I;V ), cf. Proposition 1.37, and Theorem 1.2; details areleft as an exercise, cf. also (8.118).

Remark 8.12 (The case f ∈Lp′(I;V ∗)+Lq′(I;H), p<q< +∞10). In this case, in-

stead of u ∈ W 1,p,p′(I;V, V ∗), Definition 8.1 should require u ∈ Lp(I;V )∩Lq(I;H)

with ddtu ∈ Lp′

(I;V ∗)+Lq′(I;H). The Banach space of such u’s is again embeddedinto C(I;H) and also the by-part formula (7.15) extends to hold. Then the proof ofTheorem 8.4 bears slight modifications; e.g. the estimate ‖ d

dtuτ‖Lp′(I;V ∗) ≤ C3 re-

sulting from (8.19a) is now ‖ ddtuτ‖Lp′(I;V ∗)+Lq′ (I;H) ≤ C3, the convergence (8.39c)

takes Lp′(I;V ∗) + Lq′(I;H), and the dualities 〈 d

dtu, u〉 and 〈f, u〉 refer to (1.9).

Theorem 8.13 (Weak solution). Let 1 < p ≤ q ≤ +∞, p < +∞, A : V → V ∗

be pseudomonotone11 and semicoercive such that A is weakly* continuous fromW 1,p,M (I;V, Z∗) ∩ L∞(I;H) to L∞(I;Z)∗ and satisfy (8.12) with some Z ⊂ Vdensely, let u0 ∈ H and u0τ satisfy (8.38), and let f satisfy (8.14), and (8.15a)hold, too. Then there is a weak solution u due to the Definition 8.2 and, moreover,ddtu ∈ Lq′(I;Z∗).

Proof. We test ddtuτ + A (uτ ) = fτ , which arises from (8.5) if the notation (8.6),

(8.7), and (8.8) applies, by v. Using the by-part formula (7.15), we can write

0 =⟨duτ

dt+ A (uτ )− fτ , v

⟩=⟨A (uτ )− fτ , v

⟩− ⟨dvdt

, uτ

⟩+ (v(T ), uτ (T ))− (v(0), u0τ ) (8.54)

10For this approach, we refer to Gajewski et al. [168, Chap.VI with Sect.IV.1.5].11In Theorem 8.31 we will still put off this assumption.


for any v ∈ W 1,p,p′(I;V, V ∗); as A : V → V ∗ is assumed bounded, hence A (uτ ) ∈

L∞(I;V ∗), and as also fτ ∈ L∞(I;V ∗), the dualities in (8.54) can be understoodas between Lp(I;V ) and its dual.

By using (8.17b) and (8.19b), we have the a-priori boundedness of {uτ}0<τ≤τ0

in W = W 1,p,M (I;V, Z∗)∩L∞(I;H). Thus, after choosing a subsequence, we haveuτ

∗⇀ u in W .

In view of (8.17a) and (8.18), we can select such a subsequence that alsouτ

∗⇀ u in Lp(I;V ) ∩ L∞(I;H). As in the proof of Theorem 8.9 we can see thatu = u and also that d

dtuτ∗⇀ d

dtu in M (I;Z∗) (or in Lq′(I;Z∗) if q < +∞). Asddtv ∈ Lp′

(I;V ∗) is fixed, we have 〈 ddtv, uτ 〉 → 〈 d

dtv, u〉.Now, we consider only v ∈ W 1,∞,∞(I;Z, V ∗). Then 〈fτ , v〉 → 〈f, v〉 with the

last duality between Lq′(I;Z∗) and Lq(I;Z). Also∫ T

0 〈A(uτ (t)), v(t)〉Z∗×Z dt →∫ T

0 〈A(u(t)), v(t)〉Z∗×Z dt due to the assumed weak* continuity of A . Hence (8.2)is proved at least if v(T ) = 0 = v(0). This says, in particular, that A (u) − f =− d

dtu in the sense of distributions on I. However, by (8.12), u ∈ L∞(I;H) ∩Lp(I;V ) implies A (u) ∈ Lq′(I;Z∗). By (8.14) with q ≥ p ≥ 1, also f ∈ Lq′(I;Z∗).Hencefore, d

dtu ∈ Lq′(I;Z∗) even if q = +∞.

As {uτ(T )}τ>0 is bounded in H , hence it converges (possibly as furtherselected subsequence) to some uT weakly in H . On the other hand, uτ (T ) = u0τ +∫ T

0ddtuτdt converges to u0+

∫ T

0ddtu dt = u(T ) in Z∗. Hence uT = u(T ), the further

selection was redundant, and the term (uτ (T ), v(T )) converges to (u(T ), v(T )).The convergence of (v(0), u0τ ) to (v(0), u0) is obvious.

The weak continuity of the mapping t �→ u(t) : I → H required in Def-inition 8.2 follows from its boundedness and from having information aboutddtu ∈ L1(I;Z∗), hence it is absolutely continuous as I → Z∗. �

Remark 8.14 (Semi-implicit formulae). Especially for further numerical applica-tions it is often advantageous to consider a certain “linearization” B(w, ·) : V →V ∗ of A at a current point w, likewise (but not necessarily just as) in (2.73), andthen to modify the fully implicit formula (8.5) as

ukτ − uk−1

τ

τ+B(uk−1

τ , ukτ ) = fk

τ , k ≥ 1. (8.55)

In any case, the compatibility A(u) = B(u, u) is required and linearity of B(w, ·) isan optional property from which some benefits may follow. The a-priori estimatesand convergence analysis are to be made case by case, cf. Exercises 8.74 and 8.92.Besides a linearization, semi-implicit formulae can serve to decouple systems ofequations, cf. e.g. (12.58) or Remark 8.25 and Exercise 12.28.

Remark 8.15 (Alternative estimation). Instead of the heuristics (8.20)–(8.21), onecan use


∥∥u∥∥H

d

dt

∥∥u∥∥H+ c0

∣∣u∣∣pV≤ c1

∣∣u∣∣V+ c2

∥∥u∥∥2H+⟨f, u

⟩≤ c1

∣∣u∣∣V+ c2

∥∥u∥∥2H+ C0

∥∥f1∥∥p′

V ∗+c02

∣∣u∣∣pV+(C

P

∥∥f1∥∥V ∗+∥∥f2∥∥H)∥∥u∥∥H (8.56)

with some C0 depending on c0 and p and on the Poincare constant CP. From

this, one again obtains boundedness of u in L∞(I;H) ∩ Lp(I;V ) by Gron-wall’s inequality when qualifying f = f1+f2 as in (8.14) and when using‖u‖H d

dt‖u‖H = 12

ddt‖u‖2H. Yet (8.56) suggests a modification of (8.25) and (8.27)

by using respectively the estimates 〈ukτ−uk−1

τ , ukτ 〉 ≥ ‖uk

τ‖H(‖ukτ‖H−‖uk−1

τ ‖H) and

〈fkτ , u

kτ 〉 = 〈fk

1τ+fk2τ , u

kτ 〉 ≤ C0‖fk

1τ‖p′

V ∗ + 12c0|uk

τ |pV +(CP‖fk1τ‖V ∗ + ‖fk

2τ‖H)‖ukτ‖H ,

which would turn (8.26) into the estimate

∥∥ukτ

∥∥H

‖ukτ‖H−‖uk−1

τ ‖Hτ

+c02

∣∣ukτ

∣∣pV≤ c1

∣∣ukτ

∣∣V+ c2

∥∥ukτ

∥∥2H

+ C0

∥∥fk1τ

∥∥p′

V ∗ +(C

P

∥∥fk1τ

∥∥V ∗+

∥∥fk2τ

∥∥H

)∥∥ukτ

∥∥H. (8.57)

Then, after estimating still c1∣∣uk

τ

∣∣V

≤ 14 c0∣∣uk

τ

∣∣pV

+ C′p with some constant C′pdepending on p and on c0, one can apply the Nochetto-Savare-Verdi variant of thediscrete Gronwall inequality (1.71)–(1.72) with yk := ‖uk

τ‖H , zk := 14c0|uk

τ |pV , andc = c2. One thus obtains the estimates (8.17a-c) under the restriction τ0 < 1/c2which is weaker than (8.16), and also the condition (8.15a) is redundant for thisstrategy. Having ‖uk

τ‖H estimated, one can obtain also the estimate (8.17d) againby (8.24). Alternatively to (8.8), fτ can be defined as

fτ (t) := fkτ :=

1

τ

∫ kτ

(k−1)τ

f(ϑ) dϑ for t ∈ [(k−1)τ, kτ]. (8.58)

Such fτ is called the zero-order Clement quasi-interpolant of f .12 The convergencefτ → f and thus also the condition (8.15b) can be proved.13 As for (8.29a), bythe Holder inequality, we now have more explicitly

12“Zero-order” refers to the order of polynomials used to construct fτ . For the first-orderquasi-interpolation see Remark 8.19 below. The quasi-interpolation procedure was proposed in[98].

13For a general f ∈ Lq(I;X), one can proceed as follows: Take η > 0. Using the convolu-tion with a mollifier as in (7.10) with ε > 0 small enough, we get fε ∈ C(I;X) such that‖fε−f‖Lq(I;X) ≤ η/3. As I is compact, fε : I → X is uniformly continuous and thus there

is τ0 > 0 sufficiently small such that ‖fε(t1)−fε(t2)‖X ≤ T−1/qη/3 whenever |t1−t2| ≤ τ0.

Then also ‖fε(t)−[fε]τ (t)‖X ≤ T−1/qη/3 for any 0 < τ ≤ τ0 and t ∈ I, hencefore also

‖fε−[fε]τ‖Lq(I;X) ≤ T 1/q‖fε−[fε]τ‖C(I;X) ≤ η/3. Eventually, as in (8.59), we have also

‖[fε]τ−fτ‖Lq(I;X) ≤ ‖fε−f‖Lq(I;X) ≤ η/3. Altogether, ‖fτ−f‖Lq(I;X) ≤ ‖fτ−[fε]τ‖Lq(I;X) +

‖[fε]τ−fε‖Lq(I;X) + ‖fε−f‖Lq(I;X) ≤ 13η + 1

3η + 1

3η = η for any 0 < τ ≤ τ0.


τl∑

k=1

‖fkτ ‖p

′V ∗ ≤ τ

T/τ∑k=1

‖fkτ ‖p

′V ∗ = τ

T/τ∑k=1

∥∥∥1τ

∫ kτ

(k−1)τ

f(t) dt∥∥∥p′

V ∗

≤ 1p−1√τ

T/τ∑k=1

( ∫ kτ

(k−1)τ

∥∥f(t)∥∥V ∗dt

)p′

≤T/τ∑k=1

∫ kτ

(k−1)τ

∥∥f(t)∥∥p′

V ∗dt =∥∥f∥∥p′

Lp′(I;V ∗) (8.59)

for any l = 1, . . . , k. Similar considerations hold for (8.29b), too.

8.3 Further estimates

The strategy (8.20)–(8.22) of testing the equation (8.4) by u can be modified toget better results under modified (mostly stronger) data qualification; yet, notethat the condition (8.13) on the growth of A need not be assumed in this section.This additional quality of the solution is referred to as its certain regularity. Now,besides (8.8), we can also use the approximation of f due to Clement’s quasi-interpolation (8.58).

Theorem 8.16 (Regularity). Let A : V → V ∗ be pseudomonotone and semico-ercive (8.10) with some p > 1 (its value is, in fact, not important now), and

u0 ∈ V, (8.60a)

f ∈ L2(I;H), (8.60b)

A = A1 +A2 with A1 = Φ′, Φ : V → R convex, (8.60c)

Φ(u) ≥ c0|u|qV − c1‖u‖2H , ‖A2(u)‖H ≤ C(1 + |u|q/2V +‖u‖H

)(8.60d)

for some c0, q > 0.14 Then:

(i) The Rothe sequence {uτ}τ0≥τ>0 constructed by (8.5) with fkτ from (8.58), with

u0τ = u0 and with τ0 < 12c0C

−2/max(1, 4c1T ) is bounded in W 1,∞,2(I;V,H).

(ii) Moreover, it has a weakly* convergent subsequence in this space, and if alsoV � H (a compact embedding), and

A1 : V → V ∗ is bounded and radially continuous, (8.61a)

A2 : V → V ∗ is totally continuous, (8.61b)

then every u ∈ W 1,∞,2(I;V,H) obtained as the weak* limit of a subsequence{uτ}τ>0 solves the abstract Cauchy problem (8.4).

(iii) Also, u ∈ C(I; (V,weak)).

14Often, but not necessarily, q = p with p referring to (8.10). In fact, q in only an intermediateexponent occuring in (8.60d), cf. also Exercise 8.62 below.

8.3. Further estimates 231

Let us note that, for any Banach space V2 such that V � V2 ⊂ H , thementioned weak* convergence in W 1,∞,2(I;V,H) together with the fact thatW 1,∞,2(I;V,H) ⊂ C(I; (V,weak)) implies the strong convergence uτ → u inC(I;V2); this can be seen from Lemma 7.10 for V1 = V and V3 = H .

Let us first make heuristics of the proof of (i) for a non-discretized problem:test the equation d

dtu+A(u) = f by ddtu, use 〈A(u), d

dtu〉 = ddtΦ(u)+ 〈A2(u),

ddtu〉,

which formally15 gives∥∥∥dudt

∥∥∥2H+

d

dtΦ(u) =

⟨f −A2(u),

du

dt

⟩≤ ∥∥A2(u)

∥∥2H+∥∥f(t)∥∥2

H+

1

2

∥∥∥dudt

∥∥∥2H.

(8.62)

Then we absorb the last term in the left-hand side and denote U(t) :=∫ t

0‖ ddϑu‖2Hdϑ so that ‖ d

dtu‖2H = ddtU and, by Holder’s inequality,

∥∥u(t)∥∥2H=∥∥∥u0 +

∫ t

0

du

dϑdϑ∥∥∥2H≤2(∫ t

0

∥∥∥dudϑ

∥∥∥Hdϑ)2+ 2

∥∥u0

∥∥2H

≤ 2tU(t)+2∥∥u0

∥∥2H.

(8.63)Thus

d

dt

(12U(t) + Φ(u)

)≤ ∥∥A2(u)

∥∥2H+∥∥f(t)∥∥2

H

≤ 3C2(1 + |u|qV +‖u‖2H

)+∥∥f(t)∥∥2

H

≤ 3C2(1 +

c0+c1c0

‖u‖2H +1

c0Φ(u)

)+∥∥f(t)∥∥2

H

≤ 3C2(1 + 2

c0+c1c0

(TU(t) + ‖u0‖2H

)+

1

c0Φ(u)

)+∥∥f(t)∥∥2

H. (8.64)

Then we use the Gronwall inequality. Note that it needs Φ(u0) < +∞, i.e. u0 ∈ V .Eventually, we get Φ(u(t)) + U(t) bounded independently of t∈ I, which impliesu ∈ L∞(I;V ) and ‖ d

dtu‖L2(I;H) =√U(T ) bounded.

Proof of Theorem 8.16. Multiply (8.5) by ukτ − uk−1

τ , and use⟨A(uk

τ ), ukτ − uk−1

τ

⟩=⟨Φ′(uk

τ ), ukτ − uk−1

τ

⟩+⟨A2(u

kτ ), u

kτ − uk−1

τ

⟩. (8.65)

We can estimate

〈Φ′(ukτ ), u

kτ − uk−1

τ 〉 ≥ Φ(ukτ )− Φ(uk−1

τ ) (8.66)

because Φ is convex; in fact, up to the factor 1/τ , (8.66) is a discrete analog ofthe chain rule 〈Φ′(u), d

dtu〉 = ddtΦ(u). Thus, dividing (8.65) still by τ and using

15Note that, if a-priori no other information about ddt

u than ddt

u ∈ Lp′(I; V ∗) is known, thistest cannot be rigorously made.


Young’s inequality, we get∥∥∥ukτ − uk−1

τ

τ

∥∥∥2H+

Φ(ukτ )− Φ(uk−1

τ )

τ=⟨fkτ −A2(u

kτ ),

ukτ − uk−1

τ

τ

⟩≤ ‖fk

τ ‖2H + ‖A2(ukτ )‖2H +

1

2

∥∥∥ukτ − uk−1

τ

τ

∥∥∥2H

≤ ‖fkτ ‖2H + 3C2

(1 + |uk

τ |qV +‖ukτ‖2H

)+

1

2

∥∥∥ukτ − uk−1

τ

τ

∥∥∥2H

≤ ‖fkτ ‖2H + 3C2

(1+

c0+c1c0

‖ukτ‖2H+

Φ(ukτ )

c0

)+

1

2


τ

τ

∥∥∥2H. (8.67)

We first absorb the last term in the left-hand side, and then, denoting Ukτ :=

τ−1∑k

l=1 ‖ulτ − ul−1

τ ‖2H , we further estimate

Ukτ −Uk−1

τ

2τ+Φ(uk

τ )−Φ(uk−1τ )

τ≤ ‖fk

τ ‖2H+3C2(1+2

c0+c1c0

(TUk

τ +‖u0‖2H)+Φ(uk

τ )

c0

)and use the discrete Gronwall inequality (1.68) provided τ is small enough,

namely τ < 12c0C

−2/max(1, 4c1T ), together with the estimate τ∑l

k=1 ‖f lτ‖2H =∫ lτ

0 ‖fτ‖2Hdt ≤ ∫ T

0 ‖f‖2Hdt, which is bounded independently of τ . This bounds

Φ(uτ ) in L∞(I), hence also uτ in L∞(I;V ). Also Ukτ is bounded, so in particular

we get a bound for UT/ττ = ‖ d

dtuτ‖2L2(I;H), as claimed in (i).

As to (ii), by Theorem 4.4(iv), A1 is monotone, hence for any v ∈ L∞(I;V )it holds that

0 ≤ ⟨A1(uτ )− A1(v), uτ−v⟩=⟨fτ − duτ

dt− A2(uτ )− A1(v), uτ−v

⟩. (8.68)

Let now {uτ}τ>0 refer to a selected subsequence converging weakly* inW 1,∞,2(I;V,H) and u be its limit. Then d

dtuτ ⇀ ddtu weakly in L2(I;H).

Moreover, by the Aubin-Lions lemma, W 1,∞,2(I;V,H) � L2(I;H) and there-fore uτ → u in L2(I;H). As ‖uτ − uτ‖L2(I;H) = 3−1/2τ‖ d

dtuτ‖L2(I;H) = O(τ),

cf. (8.51), also uτ → u in L2(I;H). Then limτ→0〈 ddtuτ , uτ〉 = 〈 d

dtu, u〉.16 In par-ticular,17 uτ (t) → u(t) in H for a.a. t ∈ I. As {uτ (t)}0<τ≤τ0 is bounded alsoin V , we also know that uτ (t) ⇀ u(t) weakly in V for a.a. t ∈ I. By (8.61b),〈A2(uτ (t)), uτ (t)〉 → 〈A2(u(t)), u(t)〉 for a.a. t ∈ I. By using (8.60d) and bounded-ness of {uτ}0<τ≤τ0 in L∞(I;V ), we can see that A2(uτ (t)) is bounded in H inde-pendently of τ and t. Hence 〈A2(uτ (t)), uτ (t)〉 is bounded independently of both t

and τ , hence by Lebesgue’s Theorem 1.14,∫ T

0〈A2(uτ ), uτ 〉dt →

∫ T

0〈A2(u), u〉dt.

16Alternatively, we could use the inequality (8.47) if Lemma 7.3 and the by-part integration

formula (7.22) employ the space W 1,2,2(I;H,H) instead of W 1,p,p′(I;V, V ∗).17For a moment, we can select a subsequence to guarantee this; see Theorem 1.7. When the

limit of 〈A2(uτ ), uτ 〉 is uniquely identified, we can avoid this further selection, however.


Therefore, (8.68) implies

0 ≤ limτ→0

⟨fτ−duτ

dt−A2(uτ )−A1(v), uτ−v

⟩=⟨f−du

dt−A2(u)−A1(v), u−v

⟩.

Now, we proceed byMinty’s trick by putting v := u+εz for z ∈ L∞(I;V ) arbitraryand ε > 0, which gives 〈f − d

dtu − A2(u) − A1(u + εz), εz〉 ≤ 0. Then we divideit by ε > 0, and pass ε → 0 by using (8.61a).18 As z is arbitrary, we concludef − d

dtu− A2(u)− A1(u) = 0 a.e. on I.

As to (iii), in particular we have obtained u ∈ W 1,2(I;H) ⊂ C(I;H) due toLemma 7.1. Thus, u(ϑ) → u(t) in H for ϑ → t. Since u ∈ L∞(I;V ), {u(ϑ)}ϑ∈I isbounded in V and hence (possibly up to a subsequence) u(ϑ) ⇀ v in V . Yet, sinceV ⊂ H , v = u(t). As this limit is thus determined uniquely, the whole sequence(or net) must converge to u(t) weakly. Hence, u ∈ C(I; (V,weak)). �

Remark 8.17 (Asymptotics for a special case A = Φ′ and f constant). In thisspecial case, t �→ [Φ−f ](u(t)) is non increasing because it fulfills d

dt [Φ− f ](u(t)) =

−‖ ddtu‖2H ≤ 0. Moreover, it can be shown that t �→[Φ−f ](u(t)) is convex and u(t)

tends weakly to the minimum of Φ− f for t → ∞.19

Theorem 8.18 (Regularity II). Let A : V → V ∗ be pseudomonotone and semi-coercive (8.10) with some p > 1 (whose value is again not important now), and

f ∈ W 1,2(I;V ∗), (8.69a)

u0 ∈ V such that A(u0)− f(0) ∈ H, (8.69b)⟨A(u1)−A(u2), u1 − u2

⟩ ≥ c0|u1 − u2|2V − c2‖u1 − u2‖2H (8.69c)

with some c0 > 0. Then:

(i) The Rothe sequence {uτ}τ0≥τ>0 constructed by the formula (8.5) with fkτ from

(8.58) and with u0τ = u0 is bounded in W 1,∞(I;H) ∩ W 1,2(I;V ) providedτ0 < 1/(2c2) .

(ii) Moreover, it has a weakly* convergent subsequence in this space, and if alsoV � H (a compact embedding), A = A1+A2 satisfying (8.61) with A1 mono-tone, every u ∈ W 1,∞(I;H) ∩ W 1,2(I;V ) obtained as the weak* limit of asubsequence {uτ}τ>0 solves the abstract Cauchy problem (8.4).

Heuristics of the proof of (i): apply ddt to the equation d

dtu + A(u) = f and

then test it by ddtu, use 〈 d

dtA(u),ddtu〉 ≥ c0| ddtu|2V −c2‖ d

dtu‖2H (which is a continuous

18More in detail, we proceed as in (8.165) but the common integrable majorant here is noweven in L∞(I) because we have u, z ∈ L∞(I;V ) and A1 maps bounded sets in V to boundedsets in V ∗ as assumed in (8.61a).

19More about such cases can be found, e.g., in Aubin and Cellina [28, Section 3.4] or Brezis[66, Section III.3]. See also Proposition 11.9 and Remark 8.22 below.


analog of (8.73) below). Using also (8.9), this gives

1

2

d

dt

∥∥∥dudt

∥∥∥2H+ c0

∣∣∣dudt

∣∣∣2V≤⟨ d

dt

(dudt

+A(u)),du

dt

⟩+ c2

∥∥∥dudt

∥∥∥2H

=⟨dfdt

,du

dt

⟩+ c2

∥∥∥dudt

∥∥∥2H

≤ 1

2ε

∥∥∥dfdt

∥∥∥2V ∗

+ε

2

∥∥∥dudt

∥∥∥2V+ c2

∥∥∥dudt

∥∥∥2H

≤ 1

2ε

∥∥∥dfdt

∥∥∥2V ∗

+ εC2P

∣∣∣dudt

∣∣∣2V+ εC2

P

∥∥∥dudt

∥∥∥2H+ c2

∥∥∥dudt

∥∥∥2H. (8.70)

Choosing ε < c0/C2P, (8.70) reads as

d

dt

(12

∥∥∥dudt

∥∥∥2H+(c0−εC2

P

)∫ t

0

∣∣∣dudθ

∣∣∣2Vdθ)≤ (c2+εC2

P

)∥∥∥dudt

∥∥∥2H+

1

2ε

∥∥∥dfdt

∥∥∥2V ∗

(8.71)

and then, by the Gronwall inequality, the first term gives the estimate of ddtu in

L∞(I;H) while the second one for t = T gives ddtu in L2(I;V ). Note that to apply

Gronwall’s inequality, we must have guaranteed ddtu(0)∈H , i.e. A(u0)−f(0)∈H .20

Proof of Theorem 8.18. Take (8.5) for k and for k − 1, i.e.

ukτ−uk−1

τ

τ+A(uk

τ ) = fkτ ,

uk−1τ −uk−2

τ

τ+A(uk−1

τ ) = fk−1τ , (8.72)

subtract them, then test it by ukτ − uk−1

τ , and divide it by τ2. Thus we get

1

2τ


τ

τ

∥∥∥2H− 1

2τ

∥∥∥uk−1τ −uk−2

τ

τ

∥∥∥2H+ c0

∣∣∣ukτ−uk−1

τ

τ

∣∣∣2V− c2


τ

τ

∥∥∥2H

≤⟨uk

τ−2uk−1τ +uk−2

τ

τ2,ukτ−uk−1

τ

τ

⟩+⟨A(uk

τ )−A(uk−1τ )

τ,ukτ−uk−1

τ

τ

⟩=⟨fk

τ −fk−1τ

τ,ukτ−uk−1

τ

τ

⟩≤ 1

2ε

∥∥∥fkτ −fk−1

τ

τ

∥∥∥2V ∗

+ε

2


τ

τ

∥∥∥2V

≤ 1

2ε

∥∥∥fkτ −fk−1

τ

τ

∥∥∥2V ∗

+ εC2P

∣∣∣ukτ−uk−1

τ

τ

∣∣∣2V+ εC2

P


τ

τ

∥∥∥2H; (8.73)

the first inequality is due to (8.25) for (ukτ − uk−1

τ )/τ in place of ukτ and (8.69c)

while the last one is due to the Young inequality. By extension of f for t < 0 byputting f(t) = f(0), we still have f ∈ W 1,2([−τ, T ];V ∗). Then f0

τ := f(0), and weget (u0

τ − u−1τ )/τ = f(0) − A(u0) ∈ H by the assumption.21 Absorbing the term

with ε < c0/C2Pand then summing (8.73) for k = 1, . . . , l, we obtain

1

2

∥∥∥ulτ − ul−1

τ

τ

∥∥∥2H+(c0 − εC2

P

)τ

l∑k=1

∣∣∣ukτ − uk−1

τ

τ

∣∣∣2V

≤ 1

2

∥∥f(0)−A(u0)∥∥2H+(c2 + εC2

P

)τ

l∑k=1

(∥∥∥ukτ − uk−1

τ

τ

∥∥∥2H+

1

2εdkτ

)(8.74)

20It does not mean that f(0)∈H, however. In fact, f(0) has a good sense only in V ∗.21In other words, this is the definition of uk

τ for k = −1 needed here.


where we abbreviated dkτ := ‖(fkτ −fk−1

τ )/τ‖2V ∗ . Then, provided τ ≤ τ0 < 1/(2c2),ε > 0 can be chosen so small that the discrete Gronwall inequality (1.68) applies,which gives an a-priori bound for d

dtuτ in L∞(I;H) and in L2(I;V ) provided

τ∑l

k=1 dk can be bounded independently of τ and l ≤ T/τ . This can be seen from

the estimate

τ

T/τ∑k=1

dkτ ≤ τ

T/τ∑k=1

1

τ2

∫ kτ

(k−1)τ

∫ τ

0

∥∥∥ d

dtf(t− ϑ)

∥∥∥2V ∗

dϑdt

≤ 1

τ

T/τ∑k=1

∫ τ

0

∫ kτ−ϑ

(k−1)τ−ϑ

∥∥∥ d

dtf(ξ)

∥∥∥2V ∗

dξdϑ

≤ 1

τ

T/τ∑k=1

∫ τ

0

∫ kτ

(k−2)τ

∥∥∥ d

dtf(ξ)

∥∥∥2V ∗

dξdϑ ≤ 2∥∥∥dfdt

∥∥∥2L2(I;V ∗)

(8.75)

where, for the first inequality in (8.75), we used, after the substitution t− ϑ = ξ,also

dkτ :=∥∥∥fk

τ − fk−1τ

τ

∥∥∥2V ∗

=1

τ2

∥∥∥1τ

∫ kτ

(k−1)τ

f(t)− f(t−τ) dt∥∥∥2V ∗

=1

τ4

∥∥∥ ∫ kτ

(k−1)τ

∫ τ

0

d

dtf(t−ϑ) dϑdt

∥∥∥2V ∗

≤ 1

τ4

( ∫ kτ

(k−1)τ

∫ τ

0

∥∥∥ d

dtf(t−ϑ)

∥∥∥V ∗

dϑdt)2

≤ 1

τ2

∫ kτ

(k−1)τ

∫ τ

0

∥∥∥ d

dtf(t− ϑ)

∥∥∥2V ∗

dϑdt (8.76)

where the last inequality uses Holder’s inequality.

Eventually, the convergence claimed in the point (ii) has been proved inTheorem 8.16. �

Remark 8.19 (1st-order Clement’s quasi-interpolation [98]). Defining the 1st-orderquasi-interpolant fτ ∈ W 1,∞(I;V ∗) as the piecewise affine interpolation of the

sequence {fkτ }T/τ

k=0, one can interpret (8.75) as the estimate ‖ ddtfτ‖2L2(I;V ∗) ≤

2‖ ddtf‖2L2(I;V ∗).

Remark 8.20 (Multilevel formulae). The two-level formula (8.5) is not the only op-tion to be used for theoretical investigation and for further numerical applications.An example of an alternative option is the 3-level Gear’s formula [174]:

3ukτ − 4uk−1

τ + uk−2τ

2τ+A(uk

τ ) = fkτ , k ≥ 2, (8.77)


while for k = 1 one is to use (8.5). This formula approximates the time derivativewith a higher order, may yield a better error estimate than (8.5) if a solution isenough regular, and may simultaneously have good stability properties, as shownfor a linear case in [357]. A mere convergence can be shown quite simply: use thetest by δkτ := (uk

τ − uk−1τ )/τ as in the proof of Theorem 8.16(i) and the estimate⟨3uk

τ−4uk−1τ +uk−2

τ

2τ,ukτ−uk−1

τ

τ

⟩=

3

2

∥∥δkτ∥∥2H−1

2

⟨δkτ , δ

k−1τ

⟩ ≥ ∥∥δkτ∥∥2H−1

8

∥∥δk−1τ

∥∥2H,

with the agreement that δkτ := 0 for k = 0. Summation then gives 78

∑lk=1

∥∥δkτ∥∥2H+18

∥∥δlτ∥∥2H − 18

∥∥δ1τ∥∥2H , which is to be used to modify (8.67). This gives the a-priori

estimate of uτ in W 1,2(I;H)∩L∞(I;V ) as in Theorem 8.16. The convergence canthen be proved when realizing that (8.77) can be written in the form

3

2

duτ

dt− 1

2

duRτ

dt+A (uτ ) = fτ (8.78)

with the “retarded” Rothe function uRτ defined by uR

τ (t) := uτ (t− τ) for t ∈ [τ, T ]while uR

τ (t) := uτ (t) for t ∈ [0, τ ]. Modification of the proof of Theorem 8.16(ii) isleft as an Exercise 8.58. Higher-level formulae do exist, too, and exhibit stability(and thus the a-priori estimates and convergence) but up to the level 7, i.e. atmost uk−6

τ is involved; we refer to Thomee [405, Chap.10].

Remark 8.21 (Non-autonomous case). The Rothe method can be generalized tothe time-dependent, so-called non-autonomous case (8.1); more precisely, we willconsider A : I ×V → V ∗ as a Caratheodory mapping such that the correspondingNemytskiı mapping, denoted by A := NA like (8.11), i.e.[

A (v)](t) := A

(t, v(t)

), (8.79)

satisfies (8.12). For example, A : Lp(I;V )∩L∞(I;H) → Lp′(I;V ∗) is bounded if,

instead of (8.13), the following growth condition holds:

∃ γ∈Lp′(I), C:R→R increasing :

∥∥A(t, v)∥∥V ∗ ≤ C

(‖v‖H)(γ(t)+ ‖v‖p−1V

). (8.80)

Then the Rothe sequence can be defined by the recursive formula

ukτ − uk−1

τ

τ+Ak

τ (ukτ ) = fk

τ , u0τ = u0 , with

Akτ (u) :=

1

τ

∫ kτ

(k−1)τ

A(t, u) dt , fkτ :=

1

τ

∫ kτ

(k−1)τ

f(t) dt. (8.81)

Then, e.g., τ∑l

k=1〈Akτ (u

kτ ), u

kτ 〉 =

∫ lτ

0〈A(t, uτ (t)), uτ (t)〉dt. The modification of

Lemmas 8.5 and 8.6 and Theorem 8.9 would require auxiliary smoothing like in(8.5), also Lemma 8.8 holds with its proof just straightforwardly modified, whilethe modification of Theorems 8.16 and 8.18 requires additional smoothness ofA(·, u).


Remark 8.22 (Infinite time horizon). By a subsequent continuation, one can passT → +∞ and obtain respective results on I := [0,+∞). E.g. Theorem 8.9 gives

u ∈ Lploc(I;V )∩W 1,p′

loc (I;V ∗) if f ∈ Lp′loc(I;V

∗), Theorem 8.16 gives u ∈ L∞loc(I;V )∩W 1,2

loc (I;H) if f ∈ L2loc(I;H), and Theorem 8.18 gives u ∈ W 1,∞

loc (I;H)∩W 1,2loc (I;V )

if f ∈ W 1,2loc (I;V

∗).

Remark 8.23 (Alternative estimation). One can also assume, alternatively to(8.60b), that f ∈ W 1,1(I;V ∗) in Theorem 8.16. The estimation strategy (8.62)is to be integrated over [0, t] and modified by a by-part integration and by thePoincare-type inequality (8.9) as∫ t

0

∥∥∥dudt

∥∥∥2Hdt+Φ(u(t)) = Φ(u0) +

∫ t

0

⟨f −A2(u),

du

dt

⟩dt

=⟨f(t), u(t)

⟩− ∫ t

0

⟨A2(u),

du

dt

⟩+⟨dfdt

, u⟩dt

≤ ∥∥f(t)∥∥V ∗∥∥u(t)∥∥

V+

∫ t

0

∥∥A2(u)∥∥2H

∥∥∥dudt

∥∥∥2H+∥∥∥dfdt

∥∥∥V ∗

∥∥u∥∥Vdt+Φ(u0)−

⟨f(0), u0

⟩≤ CP

∥∥f(t)∥∥V ∗

(∣∣u(t)∣∣V+∥∥u(t)∥∥

H

)+

∫ t

0

1

2

∥∥A2(u)∥∥2H+

1

2

∥∥∥dudt

∥∥∥2H

+ CP

∥∥∥dfdt

∥∥∥V ∗

(∣∣u(t)∣∣V+∥∥u(t)∥∥

H

)dt+Φ(u0)−

⟨f(0), u0

⟩≤ Cq

∥∥f(t)∥∥V ∗

(1 +

∥∥u(t)∥∥2H

)+ ε|u(t)|qV + Cq,ε

∥∥f(t)∥∥q′V ∗ +

∫ t

0

1

2

∥∥A2(u)∥∥2H

+1

2

∥∥∥dudt

∥∥∥2H+ Cq

∥∥∥dfdt

∥∥∥V ∗

(1 +

∣∣u∣∣qV+∥∥u∥∥2

H

)dt+Φ(u0)−

⟨f(0), u0

⟩(8.82)

with some Cq and Cq,ε depending on CP and q and also on ε > 0. Then, choosingε > 0 small enough and using also (8.60d) and (8.63), the strategy (8.64) modifiesas

1

2U(t) +

c0−ε

c0Φ(u(t)) ≤ Cq

∥∥f(t)∥∥V ∗

(1+∥∥u(t)∥∥2

H

)+

εc1c0

∥∥u(t)∥∥2H+ Cq,ε

∥∥f(t)∥∥q′V ∗ +

∫ t

0

1

2

∥∥A2(u)∥∥2H+

1

2

∥∥∥dudt

∥∥∥2H

+ C′q∥∥∥dfdt

∥∥∥V ∗

(1 +

1

c0Φ(u) +

c0+c1c0

‖u‖2H)dt+Φ(u0)−

⟨f(0), u0

⟩(8.83)

and, after estimating ‖u‖2H ≤ 2TU + 2‖u0‖2H , cf. (8.63), is to be finished byGronwall’s inequality. Also the qualification (8.69a) can be modified by requiringf ∈ W 1,1(I;H) and the strategy (8.70) can use the estimate⟨df

dt,du

dt

⟩≤∥∥∥dfdt

∥∥∥H

(14+∥∥∥dudt

∥∥∥2H

)(8.84)


and then treated by Gronwall’s inequality. Of course, these modifications can becombined with the original strategies so that f ∈ L2(I;H) + W 1,1(I;V ∗) andf ∈ W 1,2(I;V ∗) +W 1,1(I;H) can be considered for Theorems 8.16 and 8.18, re-spectively. Implementing these strategies into the proofs of Theorems 8.16 and8.18 needs either usage of the finer version of the discrete Gronwall inequality likein Remark 8.15 or, instead of usage (8.58), the scheme (8.5) with a controlledsmoothening of f like in (8.15). This technical difficulty related with time dis-cretisation does not occur when using the Galerkin method, as presented in thefollowing Section 8.4.

Remark 8.24 (Semiconvex potential Φ). The convexity of Φ assumed in (8.60) canbe relaxed by assuming Φ only semi-convex (with respect to the norm of H) inthe sense

∃K ∈ R+ : v �→ Φ(v) +K‖v‖2H : V → R+ is convex. (8.85)

Then, for τ sufficiently small, namely 0 < τ ≤ (2K)−2, the estimation (8.65)–(8.67) can be modified by using

⟨ukτ−uk−1

τ

τ+Φ′(uk

τ ), ukτ−uk−1

τ

⟩=⟨ uk

τ√τ+Φ′(uk

τ ), ukτ−uk−1

τ

⟩−⟨uk−1

τ√τ

, ukτ−uk−1

τ

⟩+

1−√τ

τ

∥∥ukτ−uk−1

τ

∥∥2H

≥ 1

2√τ

∥∥ukτ

∥∥2H+Φ(uk

τ )−1

2√τ

∥∥uk−1τ

∥∥2H− Φ(uk−1

τ )

−⟨uk−1

τ√τ

, ukτ−uk−1

τ

⟩+

1−√τ

τ

∥∥ukτ−uk−1

τ

∥∥2H

= Φ(ukτ )− Φ(uk−1

τ ) + τ(1−

√τ

2

)∥∥∥ukτ − uk−1

τ

τ

∥∥∥2H. (8.86)

This refines Rothe’s method and, in this respect, brings it closer to the Galerkinmethod which, when executing the estimation strategy (8.62), does not need anyconvexity of Φ at all; cf. also Exercise 8.56 below.

Remark 8.25 (Decoupling by semi-implicit Rothe method). The test by the timedifference used in Theorem 8.16 allows for interesting effects in systems. For no-tational simplicity, let us demonstrate it for a system of three equations governedby Φ(u, v, z), i.e.,

du

dt+Φ′u(u, v, z) = f1,

dv

dt+Φ′v(u, v, z) = f2,

dz

dt+Φ′z(u, v, z) = f3. (8.87)

Instead of a fully implicit discretisation, we can consider the semi-implicit scheme:


ukτ−uk−1

τ

τ+Φ′u(u

kτ , v

k−1τ , zk−1

τ ) = fk1,τ , (8.88a)

vkτ−vk−1τ

τ+Φ′v(u

kτ , v

kτ , z

k−1τ ) = fk

2,τ , (8.88b)

zkτ−zk−1τ

τ+Φ′z(u

kτ , v

kτ , z

kτ ) = fk

3,τ . (8.88c)

Testing particular equations by time differences, as in (8.65), leads to the estimates∥∥∥ukτ−uk−1

τ

τ

∥∥∥2 + Φ(ukτ , v

k−1τ , zk−1

τ )− Φ(uk−1τ , vk−1

τ , zk−1τ )

τ≤⟨fk1,τ ,

ukτ−uk−1

τ

τ

⟩,∥∥∥vkτ−vk−1

τ

τ

∥∥∥2 + Φ(ukτ , v

kτ , z

k−1τ )− Φ(uk

τ , vk−1τ , zk−1

τ )

τ≤⟨fk2,τ ,

vkτ−vk−1τ

τ

⟩,∥∥∥zkτ−zk−1

τ

τ

∥∥∥2 + Φ(ukτ , v

kτ , z

kτ )− Φ(uk

τ , vkτ , z

k−1τ )

τ≤⟨fk3,τ ,

zkτ−zk−1τ

τ

⟩(8.89)

provided Φ(·, v, z), Φ(u, ·, z), and Φ(u, v, ·) are convex. Summation then yields anotable “telescopical cancellation effect”: four terms, namely ±Φ(uk

τ , vkτ , z

k−1τ ) and

±Φ(ukτ , v

k−1τ , zk−1

τ ), mutually cancel and one obtains:∥∥∥ukτ−uk−1

τ

τ

∥∥∥2 + ∥∥∥vkτ−vk−1τ

τ

∥∥∥2 + ∥∥∥zkτ−zk−1τ

τ

∥∥∥2+

Φ(ukτ , v

kτ , z

kτ )− Φ(uk−1

τ , vk−1τ , zk−1

τ )

τ

≤⟨fk1,τ ,

ukτ−uk−1

τ

τ

⟩+⟨fk2,τ ,

vkτ−vk−1τ

τ

⟩+⟨fk3,τ ,

zkτ−zk−1τ

τ

⟩, (8.90)

and then one can proceed as in (8.67). Let us note that, in contrast to (8.87), thetime-discrete system (8.88) is decoupled and that, in contrast to a fully implicitdiscretisation and Proposition 11.6, only a separate convexity (instead of the jointconvexity) of Φ is needed now. Even more, in the spirit of Remark 8.24, onlyseparate semi-convexity of Φ is sufficient if the strategy (8.86) is applied to eachinequality in (8.89). Of course, this approach applies for an arbitrary numberof equations in (8.87); for systems of two specific differential equations cf. also[220, 252]. For f constant, denoting w = (u, v, z), A1(w) = (Φ′u(w)−f1, 0, 0),A2(w) = (0,Φ′v(w)−f2, 0), and A3(w) = (0, 0,Φ′z(w)−f3), the system (8.87) takesthe form dw

dt +A1(w)+A2(w)+A3(w) = 0 and the scheme (8.88) is revealed asa fractional-step method or also a so-called Lie-Trotter (or sequential) splittingcombined with the implicit Euler formula, cf. [142, 272, 411]:

wk−1+j/3τ − w

k−4/3+j/3τ

τ+Aj(w

k−1+j/3τ ) = 0, j = 1, 2, 3, (8.91)

where, for j=1, we put wk−1τ = (uk−1

τ , vk−1τ , zk−1

τ ). Then one finds that wk−2/3τ =

(ukτ , v

k−1τ , zk−1

τ ), wk−1/3τ = (uk

τ , vkτ , z

k−1τ ), and eventually wk

τ = (ukτ , v

kτ , z

kτ ).


8.4 Galerkin method

An alternative method to analyze evolution problems, consisting in discretizationof V , is referred to as a Faedo-Galerkin method [141], or mostly briefly as Galerkinmethod likewise in case of steady-state problems where, however, it led directly tofinite-dimensional problems. Similarly as in the proof of Theorem 2.6, we considera sequence of finite-dimensional subspaces Vk ⊂ V satisfying (2.7), i.e. Vk ⊂ Vk+1

and⋃

k∈N Vk dense in V . As also V is dense in H , for u0 ∈ H we can consider asequence {u0k}k∈N converging to u0 inH and such that u0k ∈ Vk. Now, we can verynaturally consider A also time-dependent, using the convention (8.79), even in amore general setting A : I×V → Z∗ for some Z ⊂ V densely provided

⋃k∈N Vk ⊂

Z, although mostly, in particular for the purpose of strong solutions, the caseZ = V is general enough. Then the Galerkin sequence {uk}k∈N of approximatesolutions uk ∈ W 1,p′

(I;Vk) = W 1,∞,p′(I;Vk, Vk), cf. (7.3), to the Cauchy problem

(8.1) is defined by

∀v∈Vk ∀(a.a.) t∈I :(duk

dt, v)+⟨A(t, uk(t)), v

⟩Z∗×Z

=⟨f(t), v

⟩V ∗×V

, (8.92a)

uk(0) = u0k, (8.92b)

where (·, ·) is the inner product in H as before. In fact, we have Vk ⊂ Z ⊂V⊂ H ⊂ V ∗ ⊂ Z∗, hence d

dtuk is valued also in Z∗ and it will be useful to introducethe seminorms | · |k, k ∈ N, on Z∗ defined by

|ξ|k = supv∈Vk, ‖v‖Z≤1

⟨ξ, v⟩Z∗×Z

. (8.93)

In accord with (7.5), | · |q′,k denotes the seminorm on Lq′(I;Z∗) defined by

∣∣ξ∣∣q′,k :=

(∫ T

0

∣∣ξ(t)∣∣q′kdt

)1/q′


v(t)∈Vk for a.a. t∈I

∫ T

0

⟨ξ(t), v(t)

⟩Z∗×Z

dt. (8.94)

Lemma 8.26 (Galerkin approximations, a-priori estimates). Let f ∈Lp′

(I;V ∗) + Lq′(I;H), u0 ∈H, and A : I×V → Z∗ be a Caratheodory mappingsuch that A satisfies (8.80) with p ≤ q ≤ +∞ and its restriction A : I × Z → Z∗

is semi-coercive in the sense

∃c0 > 0, c1∈Lp′(I), c2∈L1(I) ∀v ∈ Z :⟨A(t, v), v

⟩Z∗×Z

≥ c0|v|pV −c1(t)|v|V −c2(t)‖v‖2H (8.95)

with | · |V referring again to (8.9). If Vk ⊂ Z and if {u0k}k∈N is bounded in H,then there is a solution uk to (8.92) satisfying∥∥uk

∥∥L∞(I;H)∩Lp(I;V )

≤ C1 , (8.96a)∣∣∣duk

dt

∣∣∣q′,l

≤ C2 ∀k ≥ l, (8.96b)

8.4. Galerkin method 241

for any l; note that C2 does not depend on l. If, in addition, there is a selfad-joint projector Pk : H → H such that Pk(V ) = Vk and ‖Pk|Z‖L (Z,Z) is boundedindependently of k, then also ∥∥∥duk

dt

∥∥∥Lq′(I;Z∗)

≤ C3 . (8.97)

Proof. Taking a base {vki}i=1,...,nk, nk := dim(Vk), in Vk and assuming uk(t) =∑nk

i=1 cki(t)vki, (8.92) represents an initial-value problem for a system of nk ordi-nary differential equations for the coefficients (ci)i=1,...,nk

. Due to Theorem 1.44,it has a solution on some sufficiently short time interval [0, tk).

22 As the testfunctions for (8.92a) are the spaces Vk where also the approximate solution uk issought, we are authorized to put v = uk(t) in (8.92). Then, as in (8.20)–(8.21), weget the estimate:

1

2

d

dt

∥∥uk(t)∥∥2H+ c0

∣∣uk(t)∣∣pV≤ c1(t)

∣∣uk(t)∣∣V+ c2(t)

∥∥uk(t)∥∥2H+⟨f(t), uk(t)

⟩≤ Cεc1(t)

p′+ ε∣∣uk(t)

∣∣pV+ c2(t)

∥∥uk(t)∥∥2H+ CPCε

∥∥f1∥∥p′

V ∗

+CPε∣∣uk

∣∣pV+(CP

∥∥f1∥∥V ∗+∥∥f2∥∥H)(12 +

1

2

∥∥uk

∥∥2H

)(8.98)

with Cε from (1.22) and with f = f1+f2, f1 ∈ Lp′(I;V ∗), f2 ∈ Lq′(I;H). In

particular, by Gronwall’s inequality (1.66) as used in (8.22), we have an L∞(0, tk)-estimate so that uk(t) must live in a ball of Vk which is compact, and hence wecan prolong the solution on the whole interval I because, assuming the contrary,we would get a limit time inside I not allowing for any further local solution,a contradiction23. Besides, this a-priori estimate yields that uk is bounded inL∞(I;H) ∩ Lp(I;V ) independently of k, as claimed in (8.96a).

If k ≥ l, using (8.92), the estimate (8.96b) follows similarly like (8.23):

∣∣∣duk

dt

∣∣∣q′,l

= sup‖v‖Lq(I;Z)≤1v(·)∈Vl a.e.

∫ T

0

⟨f(t)−A(t, uk(t)), v(t)

⟩Z∗×Z

dt

≤ sup‖v‖Lq(I;Z)≤1

∫ T

0

⟨f(t)−A(t, uk(t)), v(t)

⟩Z∗×Z

dt =∥∥f−A (uk)

∥∥Lq′ (I;Z∗)

which is bounded due to (8.12) and the already proved estimate (8.96a), cf. also(8.33). Moreover, realizing Pkuk = uk and P ∗k = Pk and using again (8.92), and

22Note that, since A is a Caratheodory mapping, I × Rnk → Rnk : (t, c1, . . . , cnk ) �→(〈A(t,∑nk

i=1 ckivki), vkj〉)j=1,...,nk

is a Caratheodory mapping, as needed for Theorem 1.44.23In special cases, e.g. (8.80) for p ≤ 2 and C(·) bounded, the right-hand side of the underlying

system of ordinary differential equations has at most a linear growth, so the global existencefollows directly by Theorem 1.45.


also (8.12) and (8.96a), we can modify the estimate (8.33) as

⟨duk

dt, v⟩=⟨Pk

duk

dt, v⟩=⟨duk

dt, Pkv

⟩=

∫ T

0

⟨f(t)−A(t, uk(t)), Pkv(t)

⟩dt

≤(‖A (uk)‖Lq′(I;Z∗) + ‖f‖Lq′(I;Z∗)

)‖Pkv‖Lq(I;Z)

≤(N0C(C1)

(‖γ‖Lq′(I)+Cp−11

)+ ‖f‖Lq′(I;Z∗)

)∥∥Pk

∥∥L (Z,Z)

‖v‖Lq(I;Z) (8.99)

where γ and C are from (8.80) and N0 is the norm of the embedding Z ⊂ Vand thus also of the embedding V ∗ ⊂ Z∗. From this, (8.97) with C3 := (N0C(C1)(‖a‖Lq′(I)+Cp−1

1 ) + ‖f‖Lq′(I;V ∗)) supk∈N ‖Pk‖L (Z,Z) follows similarly as (8.19a).�

Remark 8.27 (The projector Pk). Taking a base {vki}i=1,...,dim(Vk) of Vk orthogonalwith respect to the inner product (·, ·) in H , by putting

Pku :=

dim(Vk)∑i=1

(u, vki)vki (8.100)

we obtain a selfadjoint projector Pk : H → H , and PkH = PkV = Vk. It remains,however, to be proved in particular cases that ‖Pk‖L (Z,Z) is bounded indepen-dently of k for a suitable Z, cf. also Remark 8.44 below.

Lemma 8.28. Let the collection {Vk}k∈N satisfy (2.7). Then⋃

k∈N L∞(I;Vk) isdense in Lp(I;V ) for any 1 ≤ p ≤ +∞.

Proof. As L∞(I;V ) is dense in Lp(I;V ), it suffices to prove it for p = +∞. Takev ∈ L∞(I;V ). As v is Bochner measurable, there is a sequence {vk}k∈N of simplefunctions such that vk(t) → v(t) for a.a. t ∈ I. Besides, the construction of vkcan be performed so that vk(I) ⊂ v(I), hence ‖vk‖L∞(I;V ) ≤ ‖v‖L∞(I;V ), andvk → v in L∞(I;V ). Now, taking vk fixed and realizing (2.7), each of the (finitenumber of) values of vk can be approximated by a value in Vl if l is sufficientlylarge, obtaining some vkl ∈ L∞(I;Vl) such that liml→∞ vkl = vk in L∞(I;V ). Thuslimk→∞ liml→∞ vkl = v and, by a suitable diagonalization, we get a sequence ofvkl attaining v. �

Lemma 8.29 (Papageorgiou [323], here generalized). Let the Caratheodorymapping A : I×V→V ∗ satisfy (8.80) and (8.95) with Z = V and q := p, and letA(t, ·) be pseudomonotone for a.a. t ∈ I. Then A is pseudomonotone on W :=W 1,p,p′

(I;V, V ∗lcs) ∩ L∞(I;H).24

Proof. Just an obvious modification of the proof of Lemma 8.8. �24Recall that V ∗

lcs denotes the dual space V ∗ considered as the locally convex space equipped

with the collection of seminorms {| · |k}k∈N which induces the seminorms on Lp′(I;V ∗) by theformula (8.94) with q := p. The pseudomonotonicity is again understood in the sense of (8.35).


Theorem 8.30 (Convergence to strong solutions). Let the assumptions ofLemmas 8.28–8.29 be fulfilled, let u0k → u0 in H with u0k ∈ Vk. Then uk ⇀ uin Lp(I;V ) (possibly in terms of subsequences) and u is a strong solution to theCauchy problem (8.1).

Proof. By (8.96a) and the reflexivity of Lp(I;V ), we can take a subsequence andsome u ∈ Lp(I;V ) such that

uk∗⇀ u in Lp(I;V ) ∩ L∞(I;H). (8.101)

Moreover, referring to the embedding Ik : Vk → V from the proof of Theorem 2.6,we have I∗l

ddtuk ⇀ ξl in any Lp′

(I;V ∗l ) and ξl+1 can be assumed as an extensionof ξl from Lp(I;Vl) to Lp(I;Vl+1).

25 By (8.96b), ‖ξl‖Lp′(I;V ∗l ) ≤ C2 independently

of l ∈ N. Hence, by density of⋃

l∈N Lp(I;Vl) in Lp(I;V ) (cf. Lemma 8.28) andby a (uniquely defined) continuous extension, we get eventually a functional u ∈Lp(I;V )∗ ∼= Lp′

(I;V ∗) whose norm can again be upper-bounded by C2. Moreover,u = d

dtu because u|Lp(I;Vl) = ξl =ddtu|Lp(I;Vl) for any l, cf. also (8.41).

Note that the initial condition u(0) = u0 is satisfied because uk(0) = u0k

and because of u0k → u0 in H and of the weak continuity of the mappingu �→ u(0) : W 1,p,p′

(I;V, V ∗lcs) → V ∗lcs by Lemma 7.1. Hence, uk(0) ⇀ u(0) inV ∗lcs. Simultaneously, uk(0) = u0k → u0 = u(0) in H ; cf. also (8.43).

For v ∈ W 1,p,p′(I;V, V ∗) let us take a sequence vk ∈ Lp(I;Vk) such that

vk → v in Lp(I;V ); such a sequence does exist due to Lemma 8.28.

From (8.92) one can see that, for any z ∈ Lp(I;Vk), one has

∫ T

0

(duk

dt, z)+⟨A(t, uk(t)), z(t)

⟩V ∗×V

dt =

∫ T

0

⟨f(t), z(t)

⟩V ∗×V

dt. (8.102)

In terms of (·, ·) as the inner product in the Hilbert space L2(I;H) and 〈·, ·〉as the duality on Lp′

(I;V ∗) × Lp(I;V ), one can rewrite (8.102) into ( ddtuk, z) +

〈A (uk), z〉 = 〈f, z〉. Putting z := vk − uk, one gets

〈A (uk), vk − uk〉 = 〈f, vk − uk〉 −(duk

dt, vk − uk

)=: I

(1)k − I

(2)k . (8.103)

As uk ⇀ u in Lp(I;V ), obviously limk→∞ I(1)k = 〈f, v − u〉.

By (8.96a), uk(T ) is bounded in H , so we can assume uk(T ) ⇀ ζ in H . By

25This is a bit technical argument: having selected a subsequence such that I∗1ddt

uk ⇀ ξ1

in Lp′(I; V ∗1 ), we can select further a subsequence such that I∗2

ddt

uk ⇀ ξ2 in any Lp′(I; V ∗2 ).

This does not violate the convergence we have already for l = 1. Then we can continue forl = 3, 4, . . . ., and eventually to make a diagonalization like in the proof of Banach Theorem 1.7,cf. Exercise 2.51.


(8.96b), we have

I∗l ζ = I∗l limk→∞

uk(T ) = limk→∞

I∗l uk(T ) = limk→∞

∫ T

0

I∗lduk

dtdt+ I∗l u0k

=

∫ T

0

ξl dt+ I∗l u0 = I∗l

(∫ T

0

du

dtdt+ u(0)

)= I∗l u(T ).

As it holds for any l ∈ N, by the density of⋃

k∈N Vk in H , we get ζ = u(T ).

Now we use (7.22) for uk ∈ W 1,p′(I;Vk).

26 Then, by the weak lower semi-continuity of ‖ · ‖2H and by using also ‖u0k‖H → ‖u0‖H and u0 = u(0), we canestimate

lim supk→∞

I(2)k = lim

k→∞

(duk

dt, vk

)− 1

2lim infk→∞

‖uk(T )‖2H +1

2limk→∞

‖u0k‖2H

≤⟨dudt

, v⟩− 1

2‖u(T )‖2H +

1

2‖u(0)‖2H =

⟨dudt

, v − u⟩, (8.104)

cf. also (8.47). Altogether,

lim infk→∞

⟨A (uk), vk − uk

⟩ ≥ ⟨f − du

dt, v − u

⟩. (8.105)

Using still the boundedness of {uk}k∈N in W and the growth assumption (8.80),we can see that {A (uk)}k∈N is bounded in Lp′

(I;V ∗) = Lp(I;V )∗. As vk → v inLp(I;V ), we have

lim supk→∞

⟨A (uk), uk − v

⟩= lim sup

k→∞

⟨A (uk), uk − vk

⟩+ lim

k→∞⟨A (uk), vk − v

⟩ ≤ ⟨dudt

− f, v − u⟩. (8.106)

In particular, for v := u we have got lim supk→∞〈A (uk), uk − u〉 ≤ 0. ByLemma 8.29, i.e. the pseudomonotonicity of A , we can conclude that

lim infk→∞

⟨A (uk), uk − v

⟩ ≥ ⟨A (u), u− v⟩

(8.107)

for any v ∈ Lp(I;V ). Joining (8.106) and (8.107), one gets 〈A (u), u−v〉 ≤〈f, u−v〉 − 〈 d

dtu, u−v〉. As it holds for v arbitrary, we can conclude that

〈A (u), v〉 = 〈f, v〉 −⟨dudt

, v⟩. (8.108)

As v is arbitrary, A (u) = f − ddtu holds a.e. on I, cf. Exercise 8.49. �

26Note that Vk ⊂ H need not be dense for it.


In the above proof, one could obtain u ∈ Lp(I;V )∗ by an alternative, al-though less constructive, argumentation: As duk

dt ∈ Lp′(I;Vk) ⊂ Lp′

(I;V ∗) ∼=Lp(I;V )∗, one can consider an extension uk ∈ Lp(I;V )∗ of duk

dt according to the

Hahn-Banach Theorem 1.5 with ‖uk‖Lp(I;V )∗ = ‖duk

dt ‖Lp′(I;Vk)≤ C2 with C2 from

(8.96b). Then, up to a subsequence, uk ⇀ u in Lp′(I;V ∗). Although in general

uk �= duk

dt , in the limit one has u = dudt . This can be seen from an analog of (8.41):

⟨u, v

⟩Lp(I;V )∗×Lp(I;V )

← ⟨uk, v

⟩Lp(I;V )∗×Lp(I;V )

=

∫ T

0

(duk

dt, v)dt

= −∫ T

0

(uk,

dv

dt

)dt → −

∫ T

0

⟨u,

dv

dt

⟩dt = −

⟨u,

dv

dt

⟩,

which holds for any v ∈ C10 (I;Vl) for a sufficiently large k, namely k ≥ l. Eventu-

ally, one uses density of⋃

l∈N C10 (I;Vl) in C1

0 (I;V ), cf. Lemma 8.28, to see that uis indeed the distributional derivative of u.

If the projectors Pk from Lemma 8.26 are at our disposal, the situation iseven simpler because, due to the estimate (8.97), one can choose directly uk = duk

dt .

Theorem 8.31 (Weak solution). Let the assumptions of Lemma 8.26 whichguarantee (8.96) be satisfied, in particular, f ∈ Lp′

(I;V ∗) + Lq′(I;H), and let Asatisfy

∃γ∈Lq′ (I), C:R→R increasing: ‖A(t, u)‖Z∗ ≤ C(‖u‖H)(γ(t)+‖u‖p/q′V

), (8.109)

with 1 < q ≤ +∞ and with some Banach space Z embedded into V densely, andinduce A weakly* continuous from W 1,p,M (I;V, Z∗) ∩ L∞(I;H) to L∞(I;Z)∗,and let u0k → u0 ∈ H. Then there is a weak solution u due to the Definition 8.2and, moreover, d

dtu ∈ Lq′(I;Z∗).

Proof. By Lemma 8.26, we have the a-priori estimate (8.96a) at our disposal,hence we choose a subsequence uk

∗⇀ u in L∞(I;H)∩Lp(I;V ). Besides, as in theproof of Theorem 8.30, d

dtu has a sense in Lq′(I;Z∗) if q < +∞ or in M (I;Z∗)if q = +∞, and d

dtuk converges to ddtu|Lq(I;Vl) in each Lq′(I;V ∗l ) if q < +∞ or in

M (I;V ∗l ) if q = +∞.Now, paraphrasing the proof of Theorem 8.13, we consider for l ≤ k fixed,

vl ∈ W 1,∞,∞(I;Vl, Z∗), put v = vl(t) into (8.92a), integrate it over [0, T ], and use

the by-part integration (7.15),27 one obtains∫ T

0

⟨A(t, uk)−f, vl

⟩− ⟨dvldt

, uk

⟩dt+

(uk(T ), vl(T )

)=(u0k, vl(0)

); (8.110)

note that (8.96a) and (8.109) guarantees A (uk) ∈ Lq′(I;Z∗). As {uk(T )}k∈N isbounded in H , hence it converges (possibly as further selected subsequence) to

27We use (7.15) with Vk instead of V , realizing that ddt

uk ∈ Lq′ (I;V ∗k ) and d

dtvl ∈ L∞(I; V ∗

k ).


some uT weakly in H . On the other hand, uk(T ) = u0k +∫ T

0ddtuk dt converges to

u0+∫ T

0ddtu dt = u(T ) in Z∗. Hence uT = u(T ), the further selection was redundant,

and limk→∞(uk(T ), vl(T )) = (u(T ), vl(T )). The convergence of limk→∞(u0k, vl(0))to (u0, vl(0)) is obvious. Using the weak* continuity of A , we can pass to the limit

in (8.110) with k → ∞, obtaining∫ T

0 〈A(t,u)−f, vl〉−〈 ddtvl, u〉dt+(u(T ), vl(T )) =

(u0, vl(0)). Taking arbitrary v ∈ W 1,∞,∞(I;Z, V ∗), by Lemma 7.2 we can considerw ∈ C1(I;Z) such that w → v in Lq(I;Z) (here we rely on q < +∞) andalso d

dt w → ddtv in Lp′

(I;V ∗). Then, e.g. by a piecewise affine interpolation andsubsequent approximation from Z to Vl, we can further approximate w by vl ∈W 1,∞(I;Vl) in W 1,p′

(I;Z). By a suitable diagonalization, passing eventually withl→∞, one gets 〈A (u)−f, vl〉L1(I;Z∗)×L∞(I;Z) → 〈A (u)−f, v〉L1(I;Z∗)×L∞(I;Z) and

also 〈 ddtvl, u〉Lp′(I;V ∗)×Lp(I;V ) → 〈 d

dtv, u〉Lp′(I;V ∗)×Lp(I;V ), so that (8.2) follows.

Moreover, it says that A (u) − f = − ddtu in the sense of distributions on I.

However, by (8.109) ensuring here (8.12), u ∈ L∞(I;H)∩Lp(I;V ) implies A (u) ∈Lq′(I;Z∗). By the assumptions of Lemma 8.26 also f ∈ Lq′(I;Z∗). Hencefore,ddtu ∈ Lq′(I;Z∗).

The weak continuity of the mapping t �→ u(t) : I → H required in Defini-tion 8.2 follows as in the proof of Theorem 8.13. �

Remark 8.32 (Monotone case: convergence via Minty’s trick28). If A(t, ·) : V → V ∗

is monotone and radially continuous, under the additional growth condition (8.80),we can use Lemma 2.9 to show that A is pseudomonotone (cf. Example 8.52)which is then employed in the proof of convergence as in Theorem 8.9 or 8.30.Alternatively, we can use Lemma 8.8 or 8.29. In this monotone case, however,these chains of arguments can be made shorter and more explicit: By the a-prioriestimates, we can select a subsequence such that

uk∗⇀ u in W 1,p,p′

(I;V, V ∗lcs) ∩ L∞(I;H). (8.111)

We use also uk(T ) ⇀ u(T ) in H and vk ∈ Lp(I;Vk), v ∈ Lp(I;V ), vk → v inLp(I;V ) as in the proof of Theorem 8.30. By (8.102), we have

0 ≤ Ik :=⟨A (uk)− A (v), uk − v

⟩=⟨A (uk), uk − vk

⟩+⟨A (uk), vk − v

⟩− ⟨A (v), uk − v⟩

=⟨f − duk

dt, uk − vk

⟩+⟨A (uk), vk − v

⟩− ⟨A (v), uk − v⟩

=1

2

∥∥u0k

∥∥2H− 1

2

∥∥uk(T )∥∥2H+⟨f, uk − vk

⟩+⟨duk

dt, vk

⟩+⟨A (uk), vk − v

⟩− ⟨A (v), uk − v⟩. (8.112)

28Cf. also the proof of Theorem 8.16(ii).

8.5. Uniqueness and continuous dependence on data 247

Using lim infk→∞ ‖uk(T )‖2H ≥ ‖u(T )‖2H as in (8.104) and using also |〈A (uk), vk−v〉| ≤ supl∈N ‖A (ul)‖Lp′(I;V ∗)‖vk − v‖Lp(I;V ) → 0, we obtain

0 ≤ lim supk→∞

Ik ≤ 1

2

∥∥u0

∥∥2H− 1

2

∥∥u(T )∥∥2H+⟨f, u− v

⟩+⟨dudt

, v⟩

− ⟨A (v), u− v⟩=⟨f − du

dt, u− v

⟩− ⟨A (v), u− v

⟩. (8.113)

Then we use the Minty-trick Lemma 2.13; put v = u+εw into (8.113), divide it byε > 0, pass ε to 0 while using the radial continuity of A ; the last argument exploitsthe radial continuity of A and the Lebesgue dominated-convergence Theorem 1.14,cf. (8.165) below.

In case A is even d-monotone and V is uniformly convex, by using (8.113)for v := u and by uniform convexity of Lp(I;V ), cf. Proposition 1.37, we get eventhe convergence uk → u in Lp(I;V ); cf. also Remark 8.11.

Remark 8.33 (Various concepts of pseudomonotonicity). There is certain free-dom in the choice of W . In general, the smaller the space W (or the finer itstopology), the bigger the collection of a-priori estimates exploited, and thus theweaker the conditions imposed on A by requiring its pseudomonotonicity asW → W ∗ by (8.35). The choice of W from Lemma 8.8 was essentially similaras in Lemma 8.29, only fitted to the particular method. We could also considerW := Lp(I;V )∩L∞(I;H) but this would enable us to treat only monotone oper-ators, cf. Example 8.52 below or Exercise 8.64 still for another W of this type. Inthe Galerkin method, smaller W (or finer topology on it) needs more difficult proofof density of Vl-valued functions in W , which can, however, be overcome by an ad-ditional condition requiring boundedness of A as a mapping into a smaller spacethan W ∗. This we indeed made in Theorem 8.30 where (8.80) implies boundednessof A : W 1,p,p′

(I;V, V ∗)→Lp′(I;V ∗) ⊂ W 1,p,p′

(I;V, V ∗)∗ and then it suffices to havean approximation in Lp(I;V ), cf. Lemma 8.28. Weakening the growth assumptionso that A is bounded as a mapping W 1,p,p′

(I;V, V ∗)→(Lp(I;V )∩L∞(I;H))∗, aswill be used in the setting of Proposition 8.39 below, would need a better approx-imation, namely in Lp(I;V ) ∩ L∞(I;H), cf. Exercise 8.55.

8.5 Uniqueness and continuous dependence on data

Weakening of concepts of solutions is always a dangerous process in the sense that,if done in a too “insensitive” way, one can loose selectivity of the definition of suchsolution: then a solution is not unique even in well qualified cases.29 Therefore,the question about uniqueness of the solution has its own theoretical importance.In addition, the analysis of uniqueness of a solution is usually closely related toanother interesting question, namely its continuous dependence on the data, i.e. awell-posedness of the problem.

29See [370] for examples of such situations.


Theorem 8.34 (Uniqueness of the strong solution). Let A satisfy, besidesassumptions guaranteeing existence of a strong solution to (8.1), also

∃c∈L1(I) ∀u, v∈V ∀(a.a.)t∈I :⟨A(t, u)−A(t, v), u−v

⟩ ≥ −c(t)‖u−v‖2H . (8.114)

Then the Cauchy problem (8.4) possesses a unique strong solution u.

Proof. Take two strong solutions u1, u2 ∈ W 1,p,p′(I;V, V ∗). Then take (8.4) for

u1 and u2 such that u1(0) = u0 = u2(0), subtract it, and put v := u1 − u2, andintegrate over (0, t). By (7.22), one gets

0 =

∫ t

0

⟨d(u1 − u2)

dt, u1 − u2

⟩dϑ+

∫ t

0

⟨A (u1)− A (u2), u1 − u2

⟩dϑ

≥ 1

2

∥∥u1(t)−u2(t)∥∥2H− 1

2

∥∥u1(0)−u2(0)∥∥2H−∫ t

0

c(ϑ)‖u1(ϑ)−u2(ϑ)‖2Hdϑ. (8.115)

Using the fact that u1(0) − u2(0) = 0 and the Gronwall inequality (1.66) withy(t) := 1

2‖u1(t)− u2(t)‖2H , C := 0, b := 0, and a(t) := c(t), we obtain y(t) ≤ 0, andtherefore u1(t)− u2(t) = 0 for any t ∈ I. �

Theorem 8.35 (Continuous dependence on f and u0). Let A satisfy assump-tions guaranteeing existence of a strong solution to (8.1) and (8.114). Then:

(i) The mapping (f, u0) �→ u : Lp′(I;V ∗)×H → W 1,p,p′

(I;V, V ∗)∩L∞(I;H),where u denotes the unique solution to the investigated problem, is(norm,weak*)-continuous, i.e. it is demicontinuous.

(ii) The mapping (f, u0) �→ u : L2(I;H)×H → C(I;H) is Lipschitz continuous.

(iii) The mapping (f, u0) �→ u : L1(I;H) ×H → C(I;H) is uniformly continuousand locally Lipschitz continuous.

(iv) Moreover, V is uniformly convex, the splitting A = A1 + A2 holds with A1

satisfying the d-monotonicity (8.53) and A2 being totally continuous as amapping W → W ∗ with W := W 1,p,p′

(I;V, V ∗) ∩ L∞(I;H). Then (f, u0) �→u : Lp′

(I;V ∗)×H → Lp(I;V ) is continuous.

Proof. As to (i), the a-priori estimates and uniqueness imply immediately theweak* convergence in W 1,p,p′

(I;V, V ∗) ∩ L∞(I;H) by paraphrasing the proof ofTheorem 8.9.

As to (ii), let us take two solutions u1, u2 ∈ W 1,p,p′(I;V, V ∗) corresponding

to two right-hand sides f1, f2 ∈ L2(I;H) and two initial conditions u01, u02 ∈ H ,abbreviate u12 := u1 − u2, f12 := f1 − f2, and u012 := u01 − u02, and then takeagain (8.4) for u1, u2, subtract it, put v := u12, and integrate over [0, t]. Likewise

8.5. Uniqueness and continuous dependence on data 249

(8.115), by (7.22) and Holder’s inequality, one gets

1

2

∥∥u12(t)∥∥2H− 1

2

∥∥u012

∥∥2H−∫ t

0

c2(ϑ)∥∥u12(ϑ)

∥∥2Hdϑ

≤∫ t

0

⟨du12

dϑ, u12(ϑ)

⟩dϑ+

∫ t

0

⟨A (u1)− A (u2), u12

⟩dϑ

=

∫ t

0

⟨f12, u12

⟩dϑ ≤

∫ t

0

1

2

∥∥f12∥∥2H +1

2

∥∥u12

∥∥2Hdϑ. (8.116)

Using the Gronwall inequality (1.66) with y(t) := ‖u12(t)‖2H , C = ‖u012‖2H , a(t) :=1 + 2c2(t), b(t) := ‖f12(t)‖2H , one gets

∥∥u12(t)∥∥2H

≤(∥∥u012

∥∥2H+

∫ t

0

∥∥f12(ϑ)∥∥2He−∫ ϑ0

1+2c2(θ)dθdϑ)

× e∫ ϑ0

1+2c2(θ)dθ ≤(∥∥u012

∥∥2H+∥∥f12∥∥2L2(I;H)

)e1+2‖c2‖L1(I) . (8.117)

As to (iii), it suffices to modify (8.116) as∫ t

0〈f12, u12〉dϑ ≤ ∫ t

0‖f12‖H(12 +

12‖u12‖2H) which allows for usage of the Gronwall inequality (1.66) with a(t) :=‖f12(t)‖H + 2c2(t) and b(t) := ‖f12(t)‖H to modify (8.117) to get ‖u12(t)‖2H ≤(‖u012‖2H + ‖f12‖L1(I;H))e

‖f12‖L1(I;H)+2‖c2‖L1(I) .To prove (iv), one can just modify (8.52) so that

1

2

∥∥u12(T )∥∥2H+⟨A1(u1)−A1(u2), u12

⟩ ≤ 1

2

∥∥u012

∥∥2H

+⟨f12, u12

⟩+⟨A2(u2)−A2(u1), u12

⟩=: I1 + I2 + I3. (8.118)

Considering u02 → u01 in H and f2 → f1 in Lp′(I;V ∗), by Step (i), we know

u2 → u1 in W weakly*. Then obviously I1 → 0 and I2 → 0. Total continuityof A2 eventually gives also I3 → 0 and d-monotonicity of A1 gives u2 → u1 inLp(I;V ). �

Now we come to uniqueness of the weak solution, which is an important as-sertion justifying Definition 8.2 whose selectivity is otherwise not entirely obvious.The serious difficulty consists in lack of regularity of the weak solution which doesnot allow for using it as a test function. Hence, we must use a suitable smoothingprocedure and the proof is much more technical than in the case of the strongsolution. Here we have at our disposal the procedure (7.18) which, unfortunately,still forces us to impose growth qualification on A corresponding to the strongsolution, so the only extension is in the right-hand side f which is not required tolive in Lp′

(I;V ∗) for Definition 8.2.

Theorem 8.36 (Uniqueness of the weak solution). Let A(t, ·) satisfy (8.114)and (8.12) with q = p and Z = V be considered. Then the weak solution accordingto Definition 8.2 is unique.


Proof. Take u1, u2 ∈ Lp(I;V ) ∩ L∞(I;H) two weak solutions, i.e. both u1 andu2 satisfy (8.2). Let us sum (8.2) for u1 and u2, smoothen u12 := u1−u2 by aregularization procedure with the properties (7.18) with considering u0 = 0 there,let us denote the result as uε

12, and then use the test function v as uε12 “continuously

cut” at some ϑ ∈ (0, T ], namely

v(t) :=

⎧⎪⎨⎪⎩uε12(t) if t ≤ ϑ,

ϑ+ε−t

εuε12(ϑ) if ϑ < t < ϑ+ ε,

0 if t ≥ ϑ+ ε.

(8.119)

This gives∫ ϑ

0

⟨A(t, u1(t))−A(t, u2(t)), u

ε12(t)

⟩− ⟨u12(t),duε

12

dt

⟩dt

+

∫ ϑ+ε

ϑ

ϑ+ε−t

ε

⟨A(t, u1(t))−A(t, u2(t)), u

ε12(ϑ)

⟩+⟨u12(t),

uε12(ϑ)

ε

⟩dt = 0.

(8.120)

By (8.12) with q = p and Z = V , we have A (ui) ∈ Lp′(I;V ∗) for

i = 1, 2. By (7.18a) and (8.114), limε→0

∫ ϑ

0〈A(t, u1(t))−A(t, u2(t)), u

ε12(t)〉dt =∫ ϑ

0〈A(t, u1(t))−A(t, u2(t)), u12(t)〉dt ≥ − ∫ ϑ

0c(t)‖u12(t)‖2Hdt. By this argument

also limε→0

∫ ϑ+ε

ϑϑ+ε−t

ε 〈A(t, u1) − A(t, u2), uε12(ϑ)〉dt = 0. We further con-

sider ϑ ∈ (0, T ] as a right Lebesgue point for u12 : I → V to guarantee

limε→0 ε−1∫ ϑ+ε

ϑ u12(t) dt = u12(ϑ), and simultaneously a left Lebesgue point for〈u∗, u12(·)〉 : I → R for any u∗ ∈H to guarantee (7.18d) at t = ϑ; here we use ageneral assumption that H and V are separable hence the set of such ϑ’s is densein I, cf. Theorem 1.35. Then, by using (7.18b-d),

lim infε→0

(−∫ ϑ

0

⟨u12,

duε12

dt

⟩dt+

∫ ϑ+ε

ϑ

⟨u12(t),

uε12(ϑ)

ε

⟩dt

)≥ lim inf

ε→0

(−∫ ϑ

0

⟨uε12,

duε12

dt

⟩dt+

⟨1ε

∫ ϑ+ε

ϑ

u12 dt, uε12(ϑ)

⟩)− lim sup

ε→0

∫ ϑ

0

⟨uε12 − u12,

duε12

dt

⟩dt

≥ lim infε→0

(12‖uε

12(0)‖2H +1

2‖uε

12(ϑ)‖2H)

+ limε→0

(⟨1ε

∫ ϑ+ε

ϑ

u12 dt, uε12(ϑ)

⟩− ‖uε

12(ϑ)‖2H)

≥ 1

2‖u12(ϑ)‖2H . (8.121)

Now we are ready to lower-bound the limit inferior of (8.120), which gives12‖u12(ϑ)‖2H − ∫ ϑ

0c(t)‖u12(t)‖2Hdt ≤ 0 for a.a. ϑ, from which u12 = 0 follows

by the Gronwall inequality (1.66). �

8.6. Application to quasilinear parabolic equations 251

8.6 Application to quasilinear parabolic equations

For Ω a bounded, Lipschitz, time-independent domain in Rn with the bound-ary Γ, we will use the notation Q := I × Ω and Σ := I × Γ and consider theinitial-boundary-value problem (with Newton-type boundary conditions) for thequasilinear parabolic 2nd-order equation:

∂u

∂t−

n∑i=1

∂

∂xiai(t, x, u,∇u) + c(t, x, u,∇u) = g(t, x) for (t, x)∈Q,

ν(x) · a(t, x, u,∇u) + b(t, x, u) = h(t, x) for (t, x)∈Σ,u(0, x) = u0(x) for x ∈ Ω,

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ (8.122)

where again ∇u := ( ∂∂x1

u, . . . , ∂∂xn

u) and ν = (ν1, . . . , νn) denotes the unitoutward normal to Γ. In accord with Convention 2.23, we occasionally omitthe arguments (t, x) in (8.122), writing shortly, e.g., ai(t, x, u,∇u) instead ofai(t, x, u(t, x),∇u(t, x)). Also, recall the notation a = (a1, . . . , an). The conven-tional setting will mostly be based on

V := W 1,p(Ω), H := L2(Ω). (8.123)

The desired reflexivity of V and the compact embedding V � H then need

p > max

(1,

2n

n+2

), (8.124)

cf. (1.34). Note that it brings no restriction on p > 1 provided n = 1 or 2, but, e.g.,for n = 3 it requires p > 6/5; cf. Remark 8.42 for the opposite case. This fits withthe abstract formulation (8.1) if A : I×W 1,p(Ω) → W 1,p(Ω)∗ and f(t) ∈ W 1,p(Ω)∗

are defined, for any v ∈ W 1,p(Ω), by⟨A(t, u), v

⟩:=

∫Ω

a(t, x, u(x),∇u(x)) · ∇v(x)

+ c(t, x, u(x),∇u(x))v(x) dx +

∫Γ

b(t, x, u(x))v(x) dS, (8.125a)

⟨f(t), v

⟩:=

∫Ω

g(t, x)v(x) dx +

∫Γ

h(t, x)v(x) dS. (8.125b)

The strong formulation of the initial-value problem (8.1) now leads to⟨∂u∂t

, v⟩W 1,p(Ω)∗×W 1,p(Ω)

+

∫Ω

a(t, x, u,∇u) · ∇v(x) + c(t, x, u,∇u)v(x) dx

+

∫Γ

b(t, x, u)v(x) dS =

∫Ω

g(t, ·)v dx+

∫Γ

h(t, ·)v dS (8.126)

for a.a. t ∈ I, and u(0, ·) = u0. Obviously, (8.126) can be obtained from (8.122)the following four steps:


1) multiplication of the first line in (8.122) by v ∈ W 1,p(Ω),

2) integration over Ω,

3) Green’s theorem in space,

4) usage of the boundary conditions in (8.122).

As such, (8.126) is called a weak formulation of (8.122) and a weak solution u isthen required to belong to W 1,p,p′

(I;W 1,p(Ω),W 1,p(Ω)∗).Alternatively, a very weak formulation (corresponding to what is on the ab-

stract level called the weak formulation, see (8.2) and Table 2 on p. 215) can beobtained by the following four steps:1) multiplication of the first line in (8.122) by v(t),

2) integration over Q,

3) Green’s theorem in space and by-part integration in time,

4) usage of the boundary and the initial conditions from (8.122).

Thus we have∫Ω

u(T, x)v(T, x) dx+

∫Q

a(t, x, u,∇u) · ∇v + c(t, x, u,∇u)v − ∂v

∂tu dxdt

+

∫Σ

b(t, x, u)v dSdt =

∫Q

gv dxdt+

∫Σ

hv dSdt+

∫Ω

u0v(0, ·) dx. (8.127)

The very weak solution u ∈ Lp(I;W 1,p(Ω)) is then to satisfy (8.127) for all v ∈W 1,∞,∞(I;W 1,∞(Ω), Lp∗′

(Ω)); here we require even ∂∂tv ∈ L∞(I;Lp∗′

(Ω)) in order

to express the duality 〈 ∂∂tv, u〉 in terms of a conventional Lebesgue integral but by

a density argument it extends for test functions used in Definition 8.2 too.In this section, we focus on the weak formulation (8.126) while the very weak

formulation (8.127) will be addressed in Section 8.7. We are to design the growthconditions on a, b, and c to guarantee the integrals in (8.126) to have a good senseand to be in L1(I) as a function of t. Let us realize that, by (1.33) and (1.63),

Lp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)) ⊂ Lp�(Q) (8.128)

for a suitable p� > p. To determine this exponent optimally, i.e. as big as possible,we use Gagliardo-Nirenberg’s Theorem 1.24 which allows here for the interpolationW 1,p(Ω) ∩ L2(Ω) ⊂ Lq(Ω), namely∥∥v∥∥

Lq(Ω)≤ C

GN

∥∥v∥∥λW 1,p(Ω)

∥∥v∥∥1−λ

L2(Ω)if

1

q≥ λ

n−p

np+

1−λ

2=:

1

q1(λ). (8.129)

Note that the function λ �→ q1(λ) is non-decreasing if W 1,p(Ω) ⊂ L2(Ω),i.e. if (8.124) holds. Then we can further estimate∥∥v∥∥q

Lq(Q)=

∫ T

0

∥∥v∥∥qLq(Ω)

dt ≤ CqGN

∫ T

0

‖v‖qλW 1,p(Ω)‖v‖q(1−λ)L2(Ω) dt

≤ CqGN

‖v‖q(1−λ)L∞(I;L2(Ω))

∫ T

0

‖v‖qλW 1,p(Ω) dt. (8.130)


The last term is bounded if qλ ≤ p, i.e. we need 1q ≥ λ

p =: 1q2(λ)

. Obviously,

the function λ �→ q2(λ) is decreasing. The aim is to choose q as big as possible,i.e. min(q1(λ), q2(λ)) as big as possible, which suggest an optimal choice for λ suchthat q1(λ) = q2(λ). A simple algebra reveals λ = n

n+2 and thus q = pn+2n ; note

that indeed λ ∈ (0, 1), as required in Theorem 1.24. Such q plays the optimal roleof the “anisotropic-interpolation” exponent p�. Thus we put

p� :=np+ 2p

n; (8.131)

cf. also [120, Sect.I.3].

To design optimally the growth conditions of the boundary terms, one needsan analog of (8.128) for the trace operator

u �→ u|Σ : Lp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)) → Lp#©(Σ) (8.132)

for a suitable p#© > p. Optimal choice of this exponent is, however, quite technical.We need a version of the Gagliardo-Nirenberg inequality generalizing (8.129) forthe fractional Sobolev-Slobodeckiı spaces. Namely, we exploit:

∥∥v∥∥Wβ,π(Ω)

≤ C′GN

∥∥v∥∥λW 1,p(Ω)

∥∥v∥∥1−λ

L2(Ω)if

1

π− β

n≥ 1

q1(λ)for π = max(2, p)

(8.133)

with q1(λ) again from (8.129). This special choice of π makes it relatively easy toshow (8.133) by interpolating the Hilbert-type Sobolev-Slobodeckiı spaces30 or byinterpolating the Sobolev/Lebesgue spaces of the same exponent p.31 Then one

30If p ≤ 2, we prove (8.133) for π = 2 by using embedding W 1,p(Ω) ⊂ W γ,2(Ω) with γ =(np+2p−2n)/(2p), and then by interpolating the Hilbert spaces W γ,2(Ω) and L2(Ω), cf. (1.44),so that we obtain

‖v∥∥Wβ,p(Ω)

≤ K1‖v‖λWγ,2(Ω)‖v‖1−λ

L2(Ω)≤ K2‖v‖β/γ

W1,p(Ω)‖v‖1−β/γ

L2(Ω)

for 0 ≤ β ≤ γ, which is just (8.133) with λ = β/γ. Cf. also [78, Lemma B.3] or [79, Lemma 2.1].31If p > 2, we prove (8.133) for π = p by interpolating W 1,p(Ω) and Lp(Ω), cf. (1.44)

for k = β, k1 = 1, k2 = 0, to obtain ‖v∥∥Wβ,p(Ω)

≤ K1‖v‖βW1,p(Ω)‖v‖1−β

Lp(Ω), and then we

interpolate Lp(Ω) in between W 1,p(Ω) and L2(Ω) by using the Gagliardo-Nirenberg inequal-

ity (1.39), i.e. ‖v‖Lp(Ω) ≤ K2‖v‖μW1,p(Ω)‖v‖1−μ

L2(Ω)with μ = q−1

1 (p) with q1 from (8.129), i.e.

μ = (2n−np)/(2n−np−2n). Thus

‖v∥∥Wβ,p(Ω)

≤ K1K1−β2 ‖v‖β+(1−β)μ

W1,p(Ω)‖v‖(1−β)(1−μ)

L2(Ω)

which is just (8.133) for λ = β+(1−β)μ; indeed, after some algebra, one can verify 1/p−β/n =1/q1(λ) for the above specified λ and μ.


can use the existence of the trace operator on Sobolev-Slobodeckiı spaces32

u �→ u|Γ : W β,π(Ω) → Lq(Γ) for q =(n−1)π

n−πβ. (8.134)

From (8.133), we have q1(λ) ≥ nπn−βπ , so that q ≤ n−1

n q1(λ). Note that both β and

π, determining an auxiliary space W β,π(Ω), have been eliminated from this esti-mate of q. Altogether, we obtained an estimate ‖v‖Lq(Γ) ≤ C‖v‖1−λ

L2(Ω)‖v‖λW 1,p(Ω).

Like (8.130), we have

∥∥v∥∥qLq(Σ)

=

∫ T

0

∥∥v∥∥qLq(Γ)

dt ≤ Cq

∫ T

0

∥∥v∥∥qλW 1,p(Ω)

∥∥v∥∥q(1−λ)

L2(Ω)dt

≤ Cq∥∥v∥∥q(1−λ)

L∞(I;L2(Ω))

∫ T

0

∥∥v∥∥qλW 1,p(Ω)

dt. (8.135)

Like in the case of (8.130), we need q ≤ q2(λ) := p/λ and our aim is to choosemin(n−1

n q1(λ), q2(λ)) as big as possible, which suggests an optimal choice for λsuch that n−1

n q1(λ) = q2(λ). From this, we obtain λ = npnp+2p−2 ; note that indeed

λ ∈ (0, 1). From this, we determine q = p/λ which plays the optimal role of the“anisotropic-interpolation” exponent p#© in (8.132). Thus we obtained

p#© :=np+ 2p− 2

nprovided p >

2n+ 2

n+ 2; (8.136)

cf. also [78, 83] for p ≤ 2 and [79] for a general case.33 To compare all introducedexponents, we have

p < p� < p∗,p < p#© < p#,

}provided (8.124) holds, (8.137a)

p < p#© < p�, (8.137b)

p < p# ≤ p∗; (8.137c)

note that the last inequality is even strict if p < n and that (8.124) is needed for thelatter inequalities in (8.137a). If p > 3n/(n+2), we have also p� < p# so that then

32There is a trace and subsequent embedding operator u �→ u|Γ : W β,π(Ω)→W β−1/π,π(Γ) ⊂Lq(Γ) for a certain q ≥ 2 sufficiently small whenever β− 1/π > 0, i.e. whenever βπ > 1; cf. [302]for π = 2 or [79] for a general π based on [408]. In analog with the exponent in Theorem 1.22, nowconsidered for β−1/π, π, and n−1 respectively in place of k, p, and n, one can easily calculateq as specified in (8.134).

33The restriction on p in (8.136) comes from q ≥ 2 needed in (8.134) for π = 2; note that itonly yields p#© > 2 and β > 1/2. This restriction is rather related with the particular ansatz used

for π in (8.133), yet the extension of p#© for lower p’s satisfying (8.124) seems to be unjustified inliterature. Anyhow, some anisotropical trace space can be identified even for small p satisfyingonly (8.124) in [78, Lemma B.3].


p < p#© < p� < p# ≤ p∗. As an example in the “physically” relevant 3-dimensionalsituation, for linear-like growth p = 2, the exponents worth remembering are:

n = 3 =⇒ 2 < 2#© =8

3< 2�=

10

3< 2#= 4 < 2∗= 6. (8.138)

The natural requirement we will assume through the following text is that

a : Q× (R×Rn) → Rn,c : Q× (R×Rn) → R,b : Σ× R → R

⎫⎬⎭ are Caratheodory mappings. (8.139)

The growth of a, c, and b fitted to (8.126) is to be designed so that the corre-

sponding Nemytskiı mappings Na, Nc, and Nb work as Lp�(Q) × Lp(Q;Rn) →

Lp′(Q;Rn), Lp�

(Q)× Lp(Q;Rn) → Lp�′(Q), and Lp#©

(Σ) → Lp#©′(Σ), respectively.

This means

∃γ∈Lp′(Q), C∈R : |a(t, x, r, s)| ≤ γ(t, x) + C|r|p�/p′

+ C|s|p−1, (8.140a)

∃γ∈Lp�′(Q), C∈R : |c(t, x, r, s)| ≤ γ(t, x) + C|r|p�−1 + C|s|p/p�′

, (8.140b)

∃γ∈Lp#©′(Σ), C∈R : |b(t, x, r)| ≤ γ(t, x) + C|r|p#©−1. (8.140c)

Lemma 8.37 (Caratheodory property of A). Let (8.139) and (8.140) be valid.Then A : I×W 1,p(Ω) → W 1,p(Ω)∗ defined by (8.125a) is a Caratheodory mapping.

Proof. Note that, as p� < p∗ and p#© < p#, (8.140) implies, in particular, thata(t, ·) : Ω × (R × Rn) → Rn, b(t, ·) : Γ × R → R and c(t, ·) : Ω × (R × Rn) → Rsatisfy the growth conditions (2.55) with ε = 0 for a.a. t ∈ I. Taking t such that(2.55) applies with ε = 0 and considering uk → u in W 1,p(Ω), we can estimate

∥∥A(t, uk)−A(t, u)∥∥W 1,p(Ω)∗ = sup

‖v‖W1,p(Ω)≤1

∫Ω

(a(t, x, uk(x),∇uk(x))

− a(t, x, u(x),∇u(x))) · ∇v(x) +

(c(t, x, uk(x),∇uk(x))

− c(t, x, u(x),∇u(x)))v(x) dx +

∫Γ

(b(t, x, uk(x)

)−b(t, x, u(x)

))v(x) dS

≤ ∥∥Na(t,·)(uk,∇uk)− Na(t,·)(u,∇u)∥∥Lp′(Ω;Rn)

+N1

∥∥Nc(t,·)(uk,∇uk)− Nc(t,·)(u,∇u)∥∥Lp∗′(Ω)

+N2

∥∥Nb(t,·)(uk)− Nb(t,·)(u)∥∥Lp#

′(Γ)

where N1 and N2 stand respectively for the norms of the embedding W 1,p(Ω) ⊂Lp∗

(Ω) and of the trace operator u �→ u|Γ : W 1,p(Ω) → Lp#

(Γ). By continuityof the Nemytskiı mappings Na(t,·), Nb(t,·), and Nc(t,·), the continuity of A(t, ·) :


W 1,p(Ω) → W 1,p(Ω)∗ follows. More specifically, here we used the embeddings andthe continuity

Lp∗(Ω)×Lp(Ω;Rn) ⊂ Lp�

(Ω)×Lp(Ω;Rn)Nc(t,·)−→ Lp�′

(Ω) ⊂ Lp∗′(Ω)

and an analogous chain for Nb(t,·):

W 1,p(Ω)trace−→ Lp#

(Γ) ⊂ Lp#©(Γ)

Nb(t,·)−→ Lp#©′(Γ) ⊂ Lp#′

(Γ).

Also, t �→ 〈A(t, u), v〉 is measurable. As W 1,p(Ω) is separable, by Pettis’ Theo-rem 1.34, A(t, ·) is also Bochner measurable. Hence A : I ×W 1,p(Ω) → W 1,p(Ω)∗

is a Caratheodory mapping, as claimed. �

For the usage of the Galerkin method, we consider a (nonspecified) se-quence of finite-dimensional subspaces Vk of W 1,p(Ω) and the respective semi-norms on W 1,p(Ω)∗ creating a locally convex topology, referred to by the notation[W 1,p(Ω)]∗lcs. We first confine ourselves to a lesser growth of the lower order terms,leading to the growth condition (8.80) and allowing for a more direct usage of theabstract theory from Sections 8.2 or 8.4.

Proposition 8.38 (Pseudomonotonicity of A ). Let the assumption (8.139)hold and a : Q× R× Rn → Rn satisfy the Leray-Lions condition

(a(t, x, r, s) − a(t, x, r, s)) · (s− s) ≥ 0, (8.141a)

(a(t, x, r, s) − a(t, x, r, s)) · (s− s) = 0 =⇒ s = s, (8.141b)

and a strengthened growth condition (8.140) hold with some ε>0 and C<+∞:

∃γ∈Lp′(Q) : |a(t, x, r, s)| ≤ γ(t, x) + C|r|(p�−ε)/p′

+ C|s|p−1, (8.142a)

∃γ∈Lp′(I;Lp∗′

(Ω)) : |c(t, x, r, s)| ≤ γ(t, x)+C|r|p�/p′+C|s|p−1, (8.142b)

∃γ∈Lp′(I;Lp#′

(Γ)) : |b(t, x, r)| ≤ γ(t, x) + C|r|p#©/p′. (8.142c)

Eventually, let the coercivity

a(t, x, r, s) · s+ c(t, x, r, s)r ≥ c0|s|p − c1(t, x)|s| − c2(t)r2, (8.143a)

b(t, x, r)r ≥ 0 (8.143b)

hold with some c0 > 0, c1 ∈Lp′(Q), and c2 ∈L1(I). Then A : W →W ∗, with W

from Lemma 8.8 or 8.29, is pseudomonotone in the sense (8.35).

Proof. The condition (8.124) implies p� ≤ p∗ and p#© ≤ p#, and therefore (8.142)guarantees that A(t, ·) : W 1,p(Ω) → W 1,p(Ω)∗ for a.a. t ∈ I.

Then we use Lemma 2.32 to show that A(t, ·) is pseudomonotone; notethat the coercivity (2.68b) is implied by (8.143) and (8.142b) similarly asin Remark 2.37. Considering the choice (8.123) together with the seminorm


|v|V := ‖∇v‖Lp(Ω;Rn), (8.143) implies the semi-coercivity assumption (8.95) withZ = V = W 1,p(Ω). Indeed, for any v ∈ W 1,p(Ω), we have⟨

A(t, v), v⟩=

∫Ω

a(v,∇v) · ∇v + c(v,∇v)v dx+

∫Γ

b(v)v dS

≥ c0∥∥∇v

∥∥pLp(Ω;Rn)

− ∥∥c1(t, ·)∥∥Lp′(Ω)

∥∥∇v∥∥Lp(Ω;Rn)

− c2(t)∥∥v∥∥2

L2(Ω)(8.144)

which verifies (8.95). Then, the inequality (8.9) just turns to be (1.55) with q = 2.We still have to verify the growth condition (8.80). As to (8.142a), we can

here, for simplicity, consider even ε = 0, i.e. (8.140a), and use an interpolation asfollows:

sup‖v‖W1,p(Ω)≤1

∫Ω

a(t, u,∇u) · ∇v dx ≤ sup‖v‖W1,p(Ω)≤1

(∥∥γ(t, ·)∥∥Lp′(Ω)

+C∥∥ |u|p�/p′∥∥

Lp′(Ω)+ C

∥∥ |∇u|p−1∥∥Lp′(Ω)

)‖∇v‖Lp(Ω;Rn)

≤ ∥∥γ(t, ·)∥∥Lp′(Ω)

+ C∥∥u∥∥p�/p′

Lp�(Ω)+ C

∥∥∇u∥∥p−1

Lp(Ω;Rn)

≤ ∥∥γ(t, ·)∥∥Lp′(Ω)

+ C∥∥u∥∥(1−λ)p�/p′

L2(Ω)

∥∥u∥∥λp�/p′

W 1,p(Ω)+ C

∥∥∇u∥∥p−1

Lp(Ω;Rn)(8.145)

provided λ = n/(n+2) as used already in the derivation of (8.131). After analgebraic manipulation we come exactly to λp�/p′ = p− 1, hence the right-handside of (8.145) turns to∥∥γ(t, ·)∥∥

Lp′(Ω)+ C

∥∥u∥∥(1−λ)p�/p′

L2(Ω)

∥∥u∥∥p−1

W 1,p(Ω)+ C

∥∥∇u∥∥p−1

Lp(Ω;Rn)

≤ max(1, C

∥∥u∥∥(1−λ)p�/p′

L2(Ω), C)(∥∥γ(t, ·)∥∥

Lp′(Ω)+∥∥∇u

∥∥p−1

Lp(Ω;Rn)

), (8.146)

which is already of the form (8.80). As to (8.142b), we estimate:


∫Ω

c(t, u,∇u)v dx ≤ sup‖v‖W1,p(Ω)≤1

(∥∥γ(t, ·)∥∥Lp∗′(Ω)

+ C∥∥ |u|p�/p′∥∥

Lp∗′(Ω)+ C

∥∥ |∇u|p−1∥∥Lp∗′(Ω)

)∥∥v∥∥Lp∗(Ω)

≤ N(∥∥γ(t, ·)∥∥

Lp∗′(Ω)+ C

∥∥u∥∥p�/p′

Lp�p∗′/p′(Ω)+ C

∥∥∇u∥∥p−1

Lp∗′(p−1)(Ω;Rn)

)≤ N

(∥∥γ(t, ·)∥∥Lp∗′(Ω)

+ CN1

∥∥u∥∥λp�/p′

W 1,p(Ω)

∥∥u∥∥(1−λ)p�/p′

L2(Ω)+ CNp−1

2

∥∥∇u∥∥p−1

Lp(Ω;Rn)

)(8.147)

with N the norm of the embedding W 1,p(Ω) ⊂ Lp∗(Ω), N1 containing the constant

from the Gagliardo-Nirenberg inequality and the norm of the embedding Lp�(Ω) ⊂

Lp�p∗′/p′(Ω), and with N2 the norm of the embedding Lp(Ω) ⊂ Lp∗′(p−1)(Ω). Using

again λ = n/(n+2), we arrive to λp�/p′ = p− 1 and thus (8.147) again complieswith the growth condition (8.80).


Analogously, the boundary term can be estimated as


∫Γ

b(u)v dS ≤ sup‖v‖W1,p(Ω)≤1

∥∥γ(t, ·) + C|u|p#©/p′∥∥Lp#

′(Γ)

∥∥v∥∥Lp#(Γ)

≤ N(∥∥γ(t, ·)∥∥

Lp#′(Γ)

+ C∥∥u∥∥p#©/p′

Lp#©

p#′/p′(Γ)

)≤ N

(∥∥γ(t, ·)∥∥Lp#

′(Γ)

+ CN1

∥∥u∥∥λp#©/p′

W 1,p(Ω)

∥∥u∥∥(1−λ)p#©/p′

L2(Ω)

)(8.148)

with N being the norm of the trace operator W 1,p(Ω) → Lp#

(Γ), and N1 con-taining the constant from the inequality (8.135) and the norm of the embedding

Lp�(Γ) ⊂ Lp#©p#′

/p′(Γ). Using λ = np/(np+2p−2) as we did for derivation of

(8.136), we arrive to λp#©/p′ = p − 1 and thus (8.148) again complies with thegrowth condition (8.80).

The pseudomonotonicity of A now follows by Lemma 8.8 or 8.29. �

For the optimal treatment of the lower-order terms, one should realize thatthe growth condition (8.80) is fitted to boundedness of A : W → Lp′

(I;W 1,p(Ω)∗)and is only sufficient for the boundedness of A : W → W ∗. This weaker bounded-ness and also the related pseudomonotonicity can be ensured by a weaker conditionthan (8.140b,c), closer to natural growth conditions (8.140):

Proposition 8.39 (Pseudomonotonicity of A : a general case). Let the as-sumptions (8.139), (8.141), (8.142a) and (8.143) hold and, with some ε > 0 andC<+∞:

∃γ∈Lp�′+ε(Q) : |c(t, x, r, s)| ≤ γ(t, x)+C|r|p�−ε−1+C|s|(p−ε)/p�′

, (8.149a)

∃γ∈Lp#©′(Σ) : |b(t, x, r)| ≤ γ(t, x)+C|r|p#©−ε−1. (8.149b)

Then A : W →W ∗, with W from Lemma 8.8 or 8.29, is pseudomonotone.

Proof. The boundedness of A : W → (Lp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)))∗ ⊂ W ∗ isto be proved by an easy combination of the growth conditions (8.142a) and (8.149)(even used with ε = 0) and the above interpolation. E.g. for the contribution of c,we can modify (8.147) to estimate:

sup‖v‖Lp(I;W1,p(Ω))≤1

‖v‖L∞(I;L2(Ω)) ≤ 1

∫Q

c(u,∇u)v dxdt ≤ sup‖v‖Lp(I;W1,p(Ω))≤1

‖v‖L∞(I;L2(Ω)) ≤ 1

(‖γ‖

Lp�′(Q)

+ C∥∥ |u|p�−1

∥∥Lp�′

(Q)+ C

∥∥ |∇u|p/p�′∥∥Lp�′

(Q)

)∥∥v∥∥Lp�(Q)

≤ sup‖v‖Lp(I;W1,p(Ω))≤1

‖v‖L∞(I;L2(Ω))≤ 1

CGN

(‖γ‖

Lp�′(Ω)

+ C∥∥u∥∥p�−1

Lp� (Ω)

+ C‖∇u‖p/p�′

Lp(Q;Rn)

)∥∥v∥∥λLp(I;W 1,p(Ω))

∥∥v∥∥1−λ

L∞(I;L2(Ω))


≤ CGN

∥∥γ∥∥Lp∗′(Ω)

+ C2GN

C∥∥u∥∥λ(p�−1)

Lp(I;W 1,p(Ω))

∥∥u∥∥(1−λ)(p�−1)

L∞(I;L2(Ω))+ CGNC

∥∥∇u∥∥p/p�′

Lp(Q;Rn)

(8.150)

with λ = n/(n+2) as used for the derivation of (8.131) and CGN from (8.130).This shows the term bounded on bounded subsets of W . Analogous calculationswork for the contribution of the boundary term b.

Then the condition (2.3b) can be proved directly for A by paraphrasingLemma 2.32, without using Lemma 8.8 or 8.29 and replacing the coercivity(8.143) by the coercivity of a(t, x, r, ·) like (2.68b). Instead of the compactness

of u �→ (u, u|Γ) : W 1,p(Ω) → Lp∗−ε(Ω) × Lp#−ε(Γ) used in Lemma 2.32, we mustuse the “interpolated” Aubin-Lions lemma 7.8 (possibly with the modification byemploying Corollary 7.9) with V2 := W 1−ε1,p(Ω), H := L2(Ω), V4 := Lq(Ω) forq−1 = 1

2 (1−λ) + λ/((p−ε1)−1−n−1), cf. (1.23). Here we use the compact embed-

ding W 1,p(Ω) � W 1−ε1,p(Ω) for any ε1 > 0, see (1.42) for the definition of theSobolev-Slobodetskiı space W 1−ε1,p(Ω). Thus we obtain W � Lp/λ(I;Lq(Ω)). Theoptimal choice of λ ∈ (0, 1) is λ = n/(n+2)−ε2, which gives that p/λ = q = p� − ε.This yields an “ε-modification” of (8.130) and thus the concrete form of (7.39) as∥∥uk−u

∥∥Lp/λ(I;Lp�−ε(Ω))

≤ CGN

∥∥uk−u∥∥1−λ

L∞(I;L2(Ω))

∥∥uk−u∥∥λLp(I;W 1−ε1,p(Ω))

→ 0

(8.151)

for any weakly* converging sequence uk∗⇀ u, which altogether yields

uk → u in Lp�−ε(Q) (8.152)

with p� from (8.131) and ε > 0 provided ε1 > 0 is sufficiently small (with respectto ε > 0). Furthermore, we can modify analogously (8.135) and claim that

uk|Σ → u|Σ in Lp#©−ε(Σ). (8.153)

Then, by the continuity of the Nemytskiı mappings Na(·,∇v) and Nb, we get

a(uk,∇v) → a(u,∇v) in Lp′(Q;Rn), cf. (8.142a), and b(uk) → b(u) in Lp#©′

(Σ).Eventually, by arguments (2.85)–(2.90) on p.52 applied on Q instead of Ω, one

obtains also c(uk,∇uk) → c(u,∇u) in Lp�′(Q).34 �

Proposition 8.40 (Existence of a weak solution). Let the assumptions of

Proposition 8.39 be valid and let g ∈ Lp′(I;Lp∗′

(Ω)), h ∈ Lp′(I;Lp#′

(Γ)), and u0 ∈L2(Ω). Then the initial-boundary-value problem (8.122) has a weak solution.

Proof. It just follows from the abstract Theorem 8.9 or 8.30 possibly (i.e. if thegrowth of b and c is indeed higher than (8.142)) based directly on Proposition 8.39instead of Lemma 8.8 or 8.29. �

34To be more precise, ε > 0 in (8.153) to be chosen small enough depending on ε > 0 in(8.149b).


Remark 8.41 (Modifications). The above Propositions 8.38–8.40 bear variousmodifications. E.g., if a(t, x, r, ·) is merely monotone (not strictly), then, as inLemma 2.32, c(t, x, r, ·) has to be affine but growth restriction (8.142b) can beslightly relaxed. Also the coercivity assumption (8.143) can be modified. E.g.,b(t, x, r)r ≥ −c3(x) − C|r|p−ε with C, ε > 0 and c3 ∈ L1(Γ) leads just toa simple modification in derivation of the above a-priori estimates. Moreover,

we can consider g ∈ L1(I;L2(Ω)), and thus also g ∈ Lp�′(Q) because, since

Lp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)) is densely embeded into Lp�(Q), also Lp�′

(Q) ⊂Lp′

(I;W 1,p(Ω)∗) +L1(I;L2(Ω))).35 Similarly, also h ∈ Lp#©′(Σ) can be considered.

Remark 8.42 (The case 1 < p ≤ 2n/(n+2)). If (8.124) does not hold, thechoice V := W 1,p(Ω) ∩ L2(Ω) and H := L2(Ω) guarantees trivially V ⊂ H .For example, the Laplacean −Δp remains semicoercive in the sense (8.10) if|v|V := ‖∇v‖Lp(Ω;Rn) is chosen. Now V �� H but V � L2−ε(Ω) for any ε > 0,which can again be used for lower-order terms through Aubin-Lions’ lemma.

Remark 8.43 (Full discretization). One can merge Rothe’s and Galerkin’s method,obtaining thus a full discretization in time and space which can be implementedat least conceptually36 on computers. Let τ > 0 be a time step and l ∈ N aspatial-discretization parameter.37 Define uk

lτ ∈ Vl ⊂ W 1,p(Ω), k = 1, . . . , T/τ , bythe following recursive formula:∫

Ω

uklτ − uk−1

lτ

τv + akτ (x, u

klτ ,∇uk

lτ ) · ∇v

+(ckτ (x, u

klτ ,∇uk

lτ )− gkτ)v dx+

∫Γ

(bkτ (x, uklτ )− τkτ )v dS = 0 (8.154)

for any v ∈ Vl, with the initial condition u0lτ = u0l where u0l ∈ Vl is defined38

by∫Ω(u0l − u0)vdx = 0 for any v ∈ Vl, and where the Clement zero-order quasi-

interpolation of the coefficients is defined by

akτ (x, r, s) :=1

τ

∫ kτ

(k−1)τ

a(t, x, r, s) dt, bkτ (x, r) :=1

τ

∫ kτ

(k−1)τ

b(t, x, r) dt, (8.155)

and analogously for ckτ . In the previous notation (8.81), we would define Akτ :

W 1,p(Ω) → W 1,p(Ω)∗ by⟨Ak

τ (u), v⟩:=

∫Ω

akτ (x, u,∇u) · ∇v + ckτ (x, u,∇u)v dx+

∫Γ

bkτ (x, u)v dS. (8.156)

35The dual space to Lp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)) contains L∞(I;L2(Ω))∗ but, in fact,

Lp�(Q) lives in a smaller space involving L1(I;L2(Ω)) instead of L∞(I;L2(Ω))∗.36At this point, various numerical-integration formulae usually have to be employed in (8.154)

and (8.155). Also, we assume that the resulting system of algebraic equations can be solvednumerically.

37With only a small loss of generality, Vl as a finite-element space with the mesh size 1/l,cf. Example 2.67.

38In other words, u0l is the L2(Ω)-orthogonal projection of u0.

8.7. Application to semilinear parabolic equations 261

Remark 8.44 (Projectors Pk). The projectors Pk(u) :=∑k

i=1

(∫Ω uvidx

)vi (cf.

(8.100)) that can alternatively be used in the abstract Galerkin method can nowemploy vi ∈ W r,2

0 (Ω) ⊂ W 1,p(Ω) (which requires r ≥ 1+n(p−2)/(2p)) solving theeigenvalue problem

Δrvi = λivi . (8.157)

Moreover, we can assume that vi makes an orthonormal basis in L2(Ω) and vi/√λi

an orthonormal basis in W r,20 (Ω). Then the projector Pk is selfadjoint, and∥∥Pk

∥∥L (L2(Ω),L2(Ω))

≤ 1 &∥∥Pk

∥∥L (W r,2

0 (Ω),W r,20 (Ω))

≤ 1 . (8.158)

The second estimate then can be used to get the a-priori estimate39∥∥∥∂uk

∂t

∥∥∥Lp′(I;W−r,2(Ω))

≤ C. (8.159)

Remark 8.45 (Pseudomonotone memory: integro-differential equations). For aCaratheodory mapping f : [Q ×Q]× R× Rn → R one can consider the nonlinearUryson integral operator (u, y) �→ (

(t, x) �→ ∫Qf(x, t, ξ, ϑ, u(ξ, ϑ), y(ξ, ϑ))dξdϑ

)which is, under certain not much restrictive conditions40, totally continuous as amapping Lp(Q;R1+n) → Lp′

(Q) and, as such, it is pseudomonotone, cf. Corol-lary 2.12. Thus one can treat e.g. the integro-differential equation

∂u

∂t− div

(|∇u|p−2∇u)+

∫Q

f(x, t, ξ, ϑ, u(ξ, ϑ),∇u(ξ, ϑ)

)dξdϑ = g. (8.160)

8.7 Application to semilinear parabolic equations

In this section we focus on the very weak formulation (8.127) in the special casewhen a(t, x, r, ·) : Rn → Rn and c(t, x, r, ·) : Rn → R are affine, i.e.

ai(t, x, r, s) :=

n∑j=1

aij(t, x, r)sj + ai0(t, x, r), i = 1, . . . , n, (8.161a)

c(t, x, r, s) :=

n∑j=1

cj(t, x, r)sj + c0(t, x, r), (8.161b)

39Unfortunately, W 1,p(Ω) is not an interpolant between L2(Ω) and W r,2(Ω) so that the in-terpolation theory to get the estimate ‖Pk‖L (W1,p(Ω),W1,p(Ω)) ≤ 1 cannot be used.

40Namely, the growth condition |f(x, t, ξ, ϑ, r, s)| ≤ γ0(x, t, ξ, ϑ) + γ1(x, t)(|r|p + |s|p) with

γ0 ∈ Lp′(Q;L1(Q)) and γ1 ∈ Lp′(Q) and the equicontinuity condition:

∀c > 0 : lim|A|→0

sup‖u‖Lp(Q)≤c‖y‖Lp(Q;Rn)≤c

∫Q

∣∣∣∣∫Af(x, t, ξ, ϑ, u(ξ, ϑ), y(ξ, ϑ)

)dξdϑ|p′

∣∣∣∣dxdt = 0.

We refer to Krasnoselskiı et al. [238, Theorem 19.3]. The latter condition is fulfilled, e.g.,if the growth condition is slightly strengthened, namely |f(x, t, ξ, ϑ, r, s)| ≤ γ0(x, t, ξ, ϑ) +γ1(x, t)(|r|p−ε + |s|p−ε) with some ε > 0 and γ0 in the form

∑finite γ0l(ξ, ϑ)γ0l(x, t) with

γ0l ∈ L1(Q) and γ0l ∈ Lp′(Q).


with aij , cj : Q×R → R Caratheodory mappings whose growth is to be designed to

induce the Nemytskiı mappings N(ai1,...,ain),N(c1,...,cn) : L2�−ε(Q) → L2(Q;Rn)

and Nai0 ,Nc0 : L2�−ε(Q) → L1(Q) with ε > 0 and with 2� := 2 + 4/n, whichcorresponds to (8.131) with p = 2. Besides, the boundary nonlinearity b : Σ×R →R is now to induce the Nemytskiı mapping Nb : L2(I;L2#−ε(Γ)) → L1(Σ). Thismeans, for i, j = 1, . . . , n,

∃γ1 ∈ L2(Q), C ∈ R : |aij(t, x, r)| ≤ γ1(t, x) + C|r|(2�−ε)/2,

|cj(t, x, r)| ≤ γ1(t, x) + C|r|(2�−ε)/2, (8.162a)

∃γ2 ∈ L1(Q), C ∈ R : |ai0(t, x, r)| ≤ γ2(t, x) + C|r|2�−ε,

|c0(t, x, r)| ≤ γ2(t, x) + C|r|2�−ε, (8.162b)

∃γ3∈L1(Σ), C ∈ R : |b(t, x, r)| ≤ γ3(t, x)+C|r|2#©−ε. (8.162c)

The exponent p = 2 is natural because the growth a(t, x, r, ·) is now linear. Notethat these requirements just guarantee that all integrals in (8.127) have a goodsense if v ∈ W 1,∞,∞(I;W 1,∞(Ω), L2∗′

(Ω)).

Lemma 8.46 (Weak continuity of A ). Let (8.161)–(8.162) hold. Then A isweakly* continuous as a mapping

W 1,2,1(I;W 1,2(Ω), [W 1,2(Ω)]∗lcs) ∩ L∞(I;L2(Ω)) → L∞(I;W 1,∞(Ω))∗.

Proof. By the Aubin-Lions lemma, we have the compact embeddingW 1,2,1(I;W 1,2(Ω), [W 1,2(Ω)]∗lcs) � L2(I;W 1−ε1,2(Ω)) for any ε1 > 0. Tak-ing ε1 > 0 suitably small, for some 0 < λ ≤ 1 we have the interpola-tion estimate ‖u‖L2�−ε(Q) ≤ C‖u‖λL2(I;W 1−ε1,2(Ω))‖u‖1−λ

L∞(I;L2(Ω)) for any u ∈L2(I;W 1−ε1,2 ∩ L∞(I;L2(Ω)); cf. also (8.151). Hence, having a weakly* conver-gent sequence {uk}k∈N in L2(I;W 1,2(Ω))∩L∞(I;L2(Ω)) with { d

dtuk}k∈N bounded

in L1(I; [W 1,2(Ω)]∗lcs), this sequence converges strongly in L2�−ε(Q). Then, by

the continuity of the Nemytskiı mappings N(ai1,...,ain),N(c1,...,cn) : L2�−ε(Q) →L2(Q;Rn) and Nai0 ,Nc0 : L2�−ε(Q) → L1(Q), it holds that∫

Q

n∑i=1

( n∑j=1

aij(uk)∂uk

∂xj+ ai0(uk)

)∂v

∂xi+

( n∑j=1

cj(uk)∂uk

∂xj+ c0(uk)

)v dxdt

→∫Q

n∑i=1

( n∑j=1

aij(u)∂u

∂xj+ ai0(u)

)∂v

∂xi+

( n∑j=1

cj(u)∂u

∂xj+ c0(u)

)v dxdt

for k → ∞ and for any v ∈ L∞(I;W 1,∞(Ω)). As in (8.153), we have now uk|Σ →u|Σ in L2#©−ε(Σ) and, by (8.162c), we have convergence also in the boundary term∫Σb(uk)v dSdt →

∫Σb(u)v dSdt. �

8.7. Application to semilinear parabolic equations 263

Proposition 8.47 (Existence of very weak solutions). Let (8.161)–(8.162)hold for some γ1 ∈ L2+ε(Q), γ2 ∈ L1+ε(Q), and γ3 ∈ L1+ε(Σ). Moreover, let

g ∈ L2(I;L2∗′(Ω)) + L1(I;L2(Ω)), h ∈ L2(I;L2#

′(Γ)), and, for some ε > 0, γ1 ∈

L2(I), γ2 ∈ L1(Q), γ3 ∈ L1(I), γ4 ∈ L1(Γ), and for a.a. (t, x)∈Q (resp. (t, x)∈Σfor (8.163)c) and all (r, s) ∈ R1+n, it holds that

n∑i=1

( n∑j=1

aij(t, x, r)sj + ai0(t, x, r))si ≥ ε|s|2 − γ1(t)|s|, (8.163a)

( n∑j=1

cj(t, x, r)sj + c0(t, x, r))r ≥ −γ2(t, x)− γ3(t)|r|2 − C|s|2−ε, (8.163b)

b(t, x, r)r ≥ −γ4(x)− C|r|2−ε. (8.163c)

Then the initial-boundary value problem (8.122) has a very weak solution.

Proof. We can use the abstract Theorem 8.31 now with V := W 1,2(Ω),Z := W 1,∞(Ω), and Vk some finite-dimensional subspaces of W 1,∞(Ω) satisfy-ing (2.7).41 The semi-coercivity (8.95) is implied by (8.163) by routine calcula-tions.42 Moreover, (8.162) implies the growth condition (8.109) with p = 2 andq < +∞, which ensures boundedness of A from L2(I;W 1,2(Ω)) ∩ L∞(I;L2(Ω))to L1+ε(I;W 1,∞(Ω)∗) with some ε > 0 (possibly different from ε in (8.162)), asrequired in Theorem 8.31. Indeed, using (8.162a,b) for simplicity heuristically withε = 0, we obtain

sup‖v‖W1,∞(Ω)≤1

∫Ω

n∑i=1

( n∑j=1

aij(u)∂u

∂xj+ ai0(u)

) ∂v

∂xidx

≤∥∥∥∥ n∑

i=1

( n∑j=1

aij(u)∂u

∂xj+ ai0(u)

) ∂v

∂xi

∥∥∥∥L1(Ω)

≤n∑

i,j=1

‖aij(u)‖L2(Ω)‖∇u‖L2(Ω;Rn) + ‖ai0(u)‖L1(Ω)

≤n∑

i,j=1

1

2‖γ1(t, ·)‖2L2(Ω)+ ‖γ2(t, ·)‖L1(Ω)+

2C+C2

2‖u‖2�

L2�(Ω)+ ‖∇u‖2L2(Ω;Rn).

Now we estimate by interpolation43 ‖u‖2�L2�(Ω)

≤ ‖u‖(1−λ)2�

L2(Ω) ‖u‖2W 1,2(Ω) for λ =

41Recall that one can consider finite-element subspaces as in Example 2.67.42We have 〈A(t, v), v〉 =

∫Ω

∑ni=1(

∑nj=1 aij(v)

∂v∂xj

+ ai0(v))∂v∂xi

+∑n

j=1 cj(v)∂v∂xj

v +

c0(v)v dx+∫Γb(v)v dS ≥ ε‖∇v‖2

L2(Ω;Rn)− 1

2measn(Ω)γ1(t)2− 1

2‖∇v‖2

L2(Ω;Rn)−‖γ2(t, ·)‖L1(Ω)−

γ3(t)‖v‖2L2 (Ω)−C‖∇v‖2−ε

L2−ε(Ω;Rn)−‖γ4‖L1(Γ) −C‖∇v‖2−ε

L2−ε(Γ)and then we can obtain (8.95)

by Young inequality.

43By (1.23), ‖u‖2�L2� (Ω)

≤ ‖u‖(1−λ)2�L2(Ω)

‖u‖λ2�L2∗ (Ω)

provided 12(1−λ) + λn−2

2n= 2�, which yields

λ = n/(n+2) and λ2� = 2 as used (8.131) for p=2.


n/(n+2), from which already the estimate of the type (8.109) follows. The contri-bution of

∑ni=1 ci(u)

∂∂xi

u + c0(u) follows from essentially the same calculations.Under the condition (8.162c), the contribution of the boundary term b is analogous,

based on the interpolation of the trace operator ‖u‖2#©L

2#©(Γ)

≤ ‖u‖(1−λ)2�

L2(Ω) ‖u‖2W 1,2(Ω)

with λ = n/(n+1) for which λ2#© = 2 as used already for (8.136). �Corollary 8.48 (Weak solutions). Let, in addition to the assumptions of Propo-sition 8.47, also g ∈ L2(I;L2∗′

(Ω)) and the growth condition (8.142) with p = 2hold. Then there is a weak solution to the initial-boundary-value problem (8.4).

Proof. It suffices to merge Proposition 8.47 and Lemma 8.4. �


This section completes the previous theory by assorted, and often physically moti-vated, examples together with some exercises accompanied mostly by brief hints.

8.8.1 General tools

Exercise 8.49. Assuming V separable, ξ∈Lp′(I;V ∗), and

∫ T

0〈ξ(t), v(t)〉V ∗×V dt = 0

for any v of the form v(t) = g(t)zi, g ∈ Lp(I), {zi}i∈N dense in V , show thatξ(t) = 0 for a.a. t ∈ I.44 Cf. Proposition 1.38.

Exercise 8.50. Modify Theorem 8.18 for c0 = 0 in (8.69c) but, on the other hand,assuming f ∈ W 1,1(I;H) in (8.69a). Only the estimate u ∈ W 1,∞(I;H) can thusbe obtained.

Exercise 8.51 (Continuous dependence on the data). Consider a sequence(fk, u0k) → (f, u0) in Lp′

(I;V ∗) × H and prove the convergence of the corre-sponding solutions as claimed in Theorem 8.35(i).

Example 8.52 (The case of A(t, ·) : V → V ∗ monotone). If A(t, ·) is monotone,radially continuous and satisfies the growth condition (8.80), then A is pseu-domonotone even as a mapping Lp(I;V ) ∩ L∞(I;H) → Lp′

(I;V ∗), i.e. no boundon the time derivative is needed. Indeed, A is obviously monotone and is boundedbecause

‖A (u)‖Lp′(I;V ∗) =(∫ T

0

‖A(t, u(t))‖p′V ∗dt

)1/p′

≤(∫ T

0

C(‖u(t)‖H)p/(p−1)

(γ(t) + ‖u(t)‖p−1

V

)p/(p−1)

dt

)(p−1)/p

≤ 21/pC(‖u(t)‖L∞(I;H)

)(‖γ‖Lp′(I) + ‖u‖p−1Lp(I;V )

)(8.164)

44Hint: Fixing zi, realize that∫ T0 〈ξ(t), v(t)〉V ∗×V dt =

∫ T0 g(t)〈ξ(t), zi〉V ∗×V dt = 0 for all g

implies 〈ξ(t), zi〉V ∗×V = 0 for a.a. t ∈ I. This holds true even if zi ranges over the countable set{zi}i∈N. As this set is dense in V , ξ(t) = 0 for a.a. t ∈ I.


where γ and C is from (8.80). Moreover, A is radially continuous because, forany u, v ∈ Lp(I;V ) ∩ L∞(I;H) and for a.a. t ∈ I, 〈A(t, u(t) + εv(t)), v(t)〉 →〈A(t, u(t)), v(t)〉 because A(t, ·) is radially continuous, and thus

⟨A (t, u+ εv), v

⟩=

∫ T

0

⟨A(t, u(t) + εv(t)), v(t)

⟩dt

→∫ T

0

⟨A(t, u(t)), v(t)

⟩dt =

⟨A (t, u), v

⟩, (8.165)

by the Lebesgue Theorem 1.14, where we used also the fact that the collection{t �→ 〈A(t, u(t) + εv(t)), v(t)〉}ε∈[0,ε0] has a common integrable majorant because,

in view of (8.80),∣∣〈A(t, u(t) + εv(t)), v(t)〉∣∣ ≤ ∥∥A(u(t) + εv(t))∥∥p′

V ∗ +∥∥v(t)∥∥p

V

≤ C(‖u+εv‖L∞(I;H)

)(γ(t) +

∥∥u(t)+εv(t)∥∥p−1

V

)p′+∥∥v(t)∥∥p

V

≤ 2p′−1C

(‖u‖L∞(I;H)+ε0‖v‖L∞(I;H)

)(γ(t)p

′+‖u(t)‖pV +εp

′0 ‖v(t)‖pV

)+∥∥v(t)∥∥p

V.

Then A is pseudomonotone by Lemma 2.9.

Example 8.53 (Totally continuous terms). Let V1 � V and A : I × V1 → V ∗ be aCaratheodory mapping satisfying (8.80) modified by replacing V with V1, i.e.

∃ γ∈Lp′(I), C:R→R increasing :

∥∥A(t, v)∥∥V ∗≤C

(‖v‖H)(γ(t)+‖v‖p−1V1

). (8.166)

Then the abstract Nemytskiı mapping A : W → Lp′(I;V ∗) with W from

Lemma 8.8 or 8.29, is totally continuous. Indeed, having a sequence uk∗⇀ u

in W , by Aubin-Lions Lemma 7.7 or its Corollary 7.9, uk → u in Lp(I;V1). Then,using

∥∥A(t, uk)∥∥V ∗≤C(γ(t)+‖uk‖p−1

V1) with C := C(supk∈N ‖uk‖L∞(I;H)) and The-

orem 1.43, we obtain A (uk) → A (u) in Lp′(I;V ∗).

Exercise 8.54. Assume A as in Example 8.52 and prove the convergence of theRothe method directly by Minty’s trick in parallel to Remark 8.32.

Exercise 8.55. Assuming (2.7) and relying upon⋃

k∈N C1(I;Vk) being dense in

W 1,p,p′(I;V, V ∗),45 prove density of

⋃k∈N L∞(I;Vk) in Lp(I;V ) ∩ L∞(I;H).46

Exercise 8.56. Consider the Galerkin approximation uk to the abstract Cauchyproblem (8.1) with data qualification (8.60), and prove the boundedness of {uk}k>0

45This density follows by Lemma 7.2 and by the famous Weierstrass theorem giving a possibilityof approaching each function C1(I;V ) by polynomials in t with coefficients in V , and eventuallyby approximating these coefficients in Vk with k sufficiently large; see e.g. Gajewski at al. [168,Sect.VI.1, Lemma 1.5] for details.

46Hint: Use approximation by C1(I;Vk) with k sufficiently large in the topology of

W 1,p,p′(I; V, V ∗), and then continuity both of the embedding W 1,p,p′(I; V, V ∗) ⊂ L∞(I;H)

by Lemma 7.3 and of the embedding W 1,p,p′(I; V, V ∗) ⊂ Lp(I;V ). Cf. also Lemma 8.28.


in W 1,2(I;H) ∩ L∞(I;V ).47 Note that, now, Φ in (8.60c,d) need not be assumedconvex.

Exercise 8.57. Consider uk as in Exercise 8.56 and the data qualification (8.69),and prove the boundedness of {uk}k>0 in W 1,∞(I;H) ∩W 1,2(I;V ).48

Exercise 8.58. Show the convergence of uτ from Gear’s formula (8.78). Modify theproof of Theorem 8.16(ii).49

Exercise 8.59. Modify Remark 8.32 for totally continuous perturbation mentionedin Example 8.53.50

Exercise 8.60. Prove the interpolation formula (1.63) by using (1.23) and Holder’sinequality.51

Exercise 8.61. Prove that all integrals in (8.126) and (8.127) have a good sense.

Exercise 8.62. Modify the estimation scenario (8.64) by requiring the (possiblynon-polynomial) growth condition ‖A2(u)‖H ≤ C(1+‖u‖H+Φ(u)1/2) instead of(8.60d), assuming (without loss of generality) that Φ ≥ 0.

8.8.2 Parabolic equation of type ∂∂tu−div(|∇u|p−2∇u)+c(u)=g

The following examples are to be considered as a detailed scrutiny of estimationtechnique on a heuristical level. Rigorously, it works if we assume that a solution uwith appropriate qualitative properties has been already obtained. Adaptation tothe Galerkin method is simple, and to the Rothe method is, in view of Sections 8.2–8.3, also quite routine.

Example 8.63 (Monotone parabolic problem: a-priori estimates). For p∈ (1,+∞)and q1, q2 ≥ 1 specified later, let us consider the initial-boundary-value problem:

∂u

∂t− div

(|∇u|p−2∇u)+ |u|q1−2u = g in Q,

|∇u|p−2 ∂u

∂ν+ |u|q2−2u = h on Σ,

u(0, ·) = u0 in Ω,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (8.167)

where g ∈ Lp′(I;Lp∗′

(Ω)) and h ∈ Lp′(I;Lp#′

(Γ)). We will prove the a-priori esti-mates on the heuristic level.

47Hint: Modify the proof of Theorem 8.16(i).48Hint: Modify the proof of Theorem 8.18(i).49Hint: Realize that d

dtuRτ − d

dtuτ ⇀ 0 in L2(I;H) due to the by-part formula 〈uR

τ −uτ ,ddt

ϕ〉 →0 for any ϕ ∈ D(I;H) because of ‖uR

τ −uτ‖L2(I;H) = O(τ) which is to be proved by a modification

of (8.50) and by using the boundedness of { ddt

uτ}0<τ≤τ0 in L2(I;H).50Hint: Generalize the proof of the “steady-state” Proposition 2.17 for the evolutionary case.51Hint: By (1.23), ‖v(·)‖Lq (Ω) ≤ ‖v(·)‖λLq1 (Ω)

‖v(·)‖1−λLq2 (Ω)

, and then integrate it over I and

use Holder’s inequality with the (mutually conjugate) exponents p1/(λp) and p2/((1−λ)p).


(1) Following the strategy (8.21) for f = f1, we use a test by u(t, ·) itself:1

2

d

dt

∥∥u∥∥2L2(Ω)

+∥∥∇u

∥∥pLp(Ω;Rn)

+∥∥u∥∥q1

Lq1 (Ω)+∥∥u∥∥q2

Lq2 (Γ)

=

∫Ω

gudx+

∫Γ

hu dS ≤ N(∥∥g∥∥

Lp∗′(Ω)+∥∥h∥∥

Lp#′(Γ)

)∥∥u∥∥W 1,p(Ω)

≤ NCP

(∥∥g∥∥Lp∗′(Ω)

+∥∥h∥∥

Lp#′(Γ)

)(∥∥∇u∥∥Lp(Ω;Rn)

+∥∥u∥∥

L2(Ω)

)≤ CεN

p′Cp′

P

(∥∥g∥∥Lp∗′(Ω)

+∥∥h∥∥

Lp#′(Γ)

)p′+ ε‖∇u‖pLp(Ω;Rn)

+NC

P

2

(∥∥g∥∥Lp∗′(Ω)

+∥∥h∥∥

Lp#′(Γ)

)(1 +

∥∥u∥∥2L2(Ω)

)(8.168)

where N is greater than the norm of the embedding/trace operator u �→ (u, u|Γ) :W 1,p(Ω) → Lp∗

(Ω) × Lp#

(Γ), and where we used the Poincare inequality in theform ‖u(t, ·)‖W 1,p(Ω) ≤ C

P(‖∇u(t, ·)‖Lp(Ω;Rn) + ‖u(t, ·)‖L2(Ω)), cf. (1.55). This,

after choosing ε < 1, using the Gronwall inequality, and integration over [0, T ],gives the a-priori estimate for u in Lp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)).

(2) The estimate for ∂∂tu in Lp′

(I;W 1,p(Ω)∗) requires assumptions on q1 andq2. In detail, imitating the scenario (8.23), we estimate:⟨∂u

∂t, v⟩=

∫Q

gv − |∇u|p−2∇u · ∇v − |u|q1−2uv dxdt

+

∫Σ

(h− |u|q2−2u

)v dSdt ≤ ∥∥∇u

∥∥p−1

Lp(Q;Rn)

∥∥∇v∥∥Lp(Q;Rn)

+∥∥ |u|q1−1

∥∥Lp′(I;Lp∗′(Ω))

∥∥v∥∥Lp(I;Lp∗(Ω))

+∥∥ |u|q2−1

∥∥Lp′(I;Lp#

′(Γ))

∥∥v∥∥Lp(I;Lp#(Γ))

+∥∥g∥∥

Lp′(I;Lp∗′(Ω))

∥∥v∥∥Lp(I;Lp∗(Ω))

+∥∥h∥∥

Lp′(I;Lp#′(Γ))

∥∥v∥∥Lp(I;Lp#(Γ))

. (8.169)

This needs q1 ≤ p and q2 ≤ p. Thus we get the estimate of ∂∂tu in Lp′

(I;W 1,p(Ω)∗).A weaker bound for q1 can be obtained by interpolation to exploit also the infor-mation u ∈ L∞(I;L2(Ω)):

∥∥ |u|q1−1∥∥Lp′(I;Lp∗′(Ω))

=

(∫ T

0

( ∫Ω

|u(t, x)|(q1−1)p∗′dx)p′/p∗′

dt

)1/p′

=∥∥u∥∥q1−1

Lp′(q1−1)(I;Lp∗′(q1−1)(Ω))≤ C

∥∥u∥∥(q1−1)λ

Lp(I;W 1,p(Ω))

∥∥u∥∥(q1−1)(1−λ)

L∞(I;L2(Ω))(8.170)

provided q1 and λ ∈ [0, 1] satisfy

1

p∗′(q1−1)≥ λ(n−p)

np+

1−λ

2and

1

p′(q1−1)≥ λ

p. (8.171)

These inequalities are upper bounds for q1. If p < 2n/(n+2), i.e. p∗ < 2, theoptimal choice of λ is simply λ = 0, and then (8.171) implies q1 ≤ 1 + 2/p∗′.


If p ≥ 2n/(n+2), the optimal choice of λ then balances both bounds for 1q1−1

occuring in (8.171), i.e. p∗′(λ(n−p)/np+ (1−λ)/2) = p′λ/p. After some algebra,for p �= n, one can see that this means λ = np(p−1)/(np2−np+2p2−2(p−n)+),and then (8.171) yields q1 ≤ (np2+2p2−2(p−n)+)/np, while for p = n a strictinequality holds. For p < n, we obtain simply q1 ≤ (np+2p)/n = p�, cf. (8.131).

The interpolation in the boundary term (hence relaxing the bound q2 ≤ p)

can be made analogously by using (8.171) with q2 and p#′in place of q1 and p∗′,

respectively.

A certain alternative approach is the estimate of ∂∂tu in a bigger space, namely

(Lp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)))∗. Then one can modify (8.169) by estimating∫Q

−|u|q1−2uv dxdt+

∫Σ

−|u|q2−2uv dSdt

≤ ∥∥ |u|q1−1∥∥Lp�′

(Q)

∥∥v∥∥Lp�(Q)

+∥∥ |u|q2−1

∥∥L

p#©′(Σ)

∥∥v∥∥L

p#©(Σ)

≤ ∥∥u∥∥q1−1

L(q1−1)p�′(Q)

∥∥v∥∥Lp�(Q)

+∥∥u∥∥q2−1

L(q2−1)p

#©′(Σ)

∥∥v∥∥L

p#©(Σ)

≤ N1

∥∥u∥∥q1−1

L(q1−1)p�′(Q)

∥∥u∥∥λ1

Lp(I;W 1,p(Ω))

∥∥u∥∥1−λ1

L∞(I;L2(Ω))

+N2

∥∥u∥∥q2−1

L(q2−1)p

#©′(Σ)

∥∥u∥∥λ2

Lp(I;W 1,p(Ω))

∥∥u∥∥1−λ2

L∞(I;L2(Ω))(8.172)

with λ1 = n/(n+2) and λ2 = np/(np+2p−2) as used for (8.130) and (8.135). The

bound of u in Lp�(Q) imposes the requirement p�′(q1−1) ≤ p�, which further

yields the restriction q1 ≤ p�, and similarly we get also q2 ≤ p#© . If p < n, thisweakens the previous restrictions on q1 and q2.

Another approach (at least for usage of the Aubin-Lions lemma) consists inweakening the dual norm to estimate ∂

∂tu in Lp′(I;W 1,p

0 (Ω)∗) ∼= Lp′(I;W−1,p′

(Ω));

note that L2(Ω) ⊂ W−1,p′(Ω) because W 1,p

0 (Ω) ⊂ L2(Ω) densely. For such anestimate one takes v in (8.169) from Lp(I;W 1,p

0 (Ω)) so that the term with q2completely vanishes, hence no restriction on q2 is imposed for this estimate.

(3) To make a test by v := ∂∂tu(t, ·), we assume g ∈ L2(Q) and h = 0. Then

∥∥∥∂u∂t

∥∥∥2L2(Ω)

+1

p

d

dt

∥∥∇u∥∥pLp(Ω;Rn)

+1

q1

d

dt

∥∥u∥∥q1Lq1 (Ω)

+1

q2

d

dt

∥∥u∥∥q2Lq2(Γ)

=

∫Ω

g(t, ·)∂u∂t

dx ≤ 1

2

∥∥g∥∥2L2(Ω)

+1

2

∥∥∥∂u∂t

∥∥∥2L2(Ω)

. (8.173)

Assuming u0 ∈ W 1,p(Ω)∩L2(Ω), which means u0 ∈ W 1,p(Ω) if p satisfies (8.124),by the Gronwall inequality, we thus get the estimate for u in L∞(I;W 1,p(Ω) ∩Lq1(Ω)) ∩W 1,2(I;L2(Ω)) and for u|Σ in L∞(I;Lq2(Γ)) for q1, q2 ≥ 1 arbitrary.

Alternatively, one can assume g ∈ W 1,1(I;Lp∗′(Ω)) and h ∈ W 1,1(I;Lp#′

(Γ))


and use the strategy from Remark 8.23. Then we can estimate∫ t

0

∫Ω

g∂u

∂tdxdt =

∫Ω

g(t, x)u(t, x)− g(0, x)u0(x) dx −∫ t

0

∫Ω

u∂g

∂tdxdt

≤ ∥∥g(t)∥∥Lp∗′(Ω)

∥∥u(t)∥∥Lp∗ (Ω)

+

∫ t

0

∥∥u∥∥Lp∗(Ω)

∥∥∥∂g∂t

∥∥∥Lp∗′(Ω)

dt+

∫Ω

g(0)u0 dx

≤ C∥∥g(t)∥∥

Lp∗′(Ω)


+∥∥u∥∥

L2(Ω)

)+

∫ t

0

C(∥∥∇u

∥∥Lp(Ω;Rn)

+∥∥u∥∥

Lq1(Ω)

)∥∥∥∂h∂t

∥∥∥Lp#

′(Γ)

dt+

∫Ω

g(0)u0 dx

≤ C∥∥g(t)∥∥

Lp∗′(Ω)

∥∥∇u∥∥Lp(Ω;Rn)

+ C2∥∥g(t)∥∥2

Lp∗′(Ω)+∥∥u∥∥2

L2(Ω)

+

∫ t

0

C′(1 +


+∥∥u∥∥q1

Lq1 (Ω)

)∥∥∥∂h∂t

∥∥∥Lp#

′(Γ)

dt+

∫Ω

g(0)u0 dx

and then proceed by Gronwall’s inequality, and similarly we can use∫ t

0

∫Γ

h∂u

∂tdSdt =

∫Γ

h(t, x)u(t, x) − h(0, x)u0(x) dS −∫ t

0

∫Γ

u∂h

∂tdSdt

≤ ∥∥h(t)∥∥Lp#

′(Γ)

∥∥u(t)∥∥Lp#(Γ)

+

∫ t

0

∥∥u∥∥Lp# (Γ)

∥∥∥∂h∂t

∥∥∥Lp#

′(Γ)

dt+

∫Γ

h(0)u0 dS

≤ C∥∥h(t)∥∥

Lp#′(Γ)


+∥∥u∥∥

L2(Ω)

)+

∫ t

0

C(∥∥∇u

∥∥Lp(Ω;Rn)

+∥∥u∥∥

Lq1 (Ω)

)∥∥∥∂h∂t

∥∥∥Lp#

′(Γ)

dt+

∫Γ

h(0)u0 dS

≤ C∥∥h(t)∥∥

Lp#′(Γ)

∥∥∇u∥∥Lp(Ω;Rn)

+ C2∥∥h(t)∥∥2

Lp#′(Γ)

+∥∥u∥∥2

L2(Ω)

+

∫ t

0

C′(1 +


+∥∥u∥∥q1

Lq1(Ω)

)∥∥∥∂h∂t

∥∥∥Lp#

′(Γ)

dt+

∫Γ

h(0)u0 dS.

(4) Further, we apply ∂∂t to the weak formulation of the equation with the

boundary conditions in (8.167), then use the test function v = ∂∂tu, and estimate

∂

∂t

(|∇u|p−2∇u) · ∇∂u

∂t= |∇u|p−2 ∂∇u

∂t· ∂∇u

∂t

+((p−2)|∇u|p−4∇u·∂∇u

∂t

)(∇u·∂∇u

∂t

)= |∇u|p−2

∣∣∣∂∇u

∂t

∣∣∣2 + p−2

4|∇u|p−4

(∂|∇u|2∂t

)2≥ 4

p2

(∂|∇u|p/2∂t

)2+

p−2

4

((|∇u|2)(p−4)/4 ∂|∇u|2∂t

)2=( 4

p2+

4p−8

p2

)(∂|∇u|p/2∂t

)2≥ 0 (8.174)


if p ≥ 1. Similar calculations work for the lower-order terms when “forgetting”∇’s for q1 ≥ 1 and q2 ≥ 1.52 Altogether, one gets

1

2

d

dt

∥∥∥∂u∂t

∥∥∥2L2(Ω)

+4p−4

p2

∥∥∥∂|∇u|p/2∂t

∥∥∥2L2(Ω)

+4q1−4

q21

∥∥∥∂|u|q1/2∂t

∥∥∥2L2(Ω)

+4q2−4

q22

∥∥∥∂|u|q2/2∂t

∥∥∥2L2(Γ)

≤∫Ω

∂g

∂t

∂u

∂tdx+

∫Γ

∂h

∂t

∂u

∂tdS =: I1(t) + I2(t). (8.175)

The integral I1 can be estimated as ‖ ∂∂tg‖L2(Ω)(

14+‖ ∂

∂tu‖2L2(Ω)), cf. (8.84). It needs∂g∂t ∈ L1(I;L2(Ω)). Alternatively, we can estimate I1 integrated over [0, t] as

∫ t

0

I1(t) dt =

∫ t

0

∫Ω

∂g

∂t

∂u

∂tdxdt =

∫Ω

∂g

∂t(t, x)u(t, x) dx

−∫ t

0

∫Ω

∂2g

∂ϑ2(ϑ, x)u(ϑ, x) dxdϑ −

∫Ω

∂g

∂t(0, x)u0(x) dx (8.176)

which is bounded if g ∈ W 2,1(I;Lp∗′(Ω)), when the estimate of u in

L∞(I;W 1,p(Ω)) obtained already at Step (3) is employed. Similarly, the integral∫ t

0 I2(t) dt is to be treated by

∫ t

0

I2(t) dt =

∫ t

0

∫Γ

∂h

∂t

∂u

∂tdSdt =

∫Γ

∂h

∂t(t, x)u(t, x) dS

−∫ t

0

∫Γ

∂2h

∂ϑ2(ϑ, x)u(ϑ, x) dSdϑ −

∫Γ

∂h

∂t(0, x)u0(x) dS (8.177)

which is bounded if h ∈ W 2,1(I;Lp#′(Γ)). Then, usage of the Gronwall in-

equality requires g ∈ W 1,2(I;L2(Ω)), and u0 ∈ W 2,q(Ω) ∩ L2(q1−1)(Ω) withq ≥ 2p∗/(p∗ − 2p + 4),53 and it gives the estimate u in W 1,∞(I;L2(Ω)) and of|∇u|p/2 in W 1,2(I;L2(Ω)) ⊂ L∞(I;L2(Ω)), which yields u ∈ L∞(I;W 1,p(Ω)).

If p ≥ 2, the term 4p−4p2 (∂|∇u|p/2

∂t )2 = (p−1)|∇u|p−2| ∂∂t∇u|2 in (8.174) gives,

through (1.46), an estimate of ∇u in the fractional space Lp(Ω;W 2/p−ε,p(I;Rn))∼= W 2/p−ε,p(I;Lp(Ω;Rn)).

52Note that (8.174) then allows for a modification ∂∂t

(|u|q−2u) ∂∂t

u = (q − 1)|u|q−2( ∂∂t

u)2 ≥ 0for both q = q1 ≥ 1 and q = q2 ≥ 1.

53This condition implies Δpu0 ∈ L2(Ω), cf. also (8.69b), because of the obvious estimate∫Ω |Δpv|2dx ≤ (p−1)2

∫Ω |∇v|2p−4|∇2v|2dx ≤ (p−1)2‖∇v‖2p−4

L(2p−4)q′ (Ω;Rn)‖∇2v‖2

L2q(Ω;Rn×n)≤

N‖v‖2(p−1)

W2,q (Ω), cf. (2.139).


For p = 2, the term I1 can be estimated more finely as

I1(t) ≤∥∥∥∂g∂t

∥∥∥L2∗′(Ω)

∥∥∥∂u∂t

∥∥∥L2∗ (Ω)

≤ 1

4ε

∥∥∥∂g∂t

∥∥∥2L2∗′(Ω)

+ ε∥∥∥∂u∂t

∥∥∥2L2∗ (Ω)

≤ 1

4ε

∥∥∥∂g∂t

∥∥∥2L2∗′(Ω)

+N21 ε∥∥∥∂u∂t

∥∥∥2L2(Ω)

+N21 ε∥∥∥∇∂u

∂t

∥∥∥2L2(Ω;Rn)

(8.178)

where N1 is the norm of the embedding W 1,2(Ω) ⊂ L2∗(Ω). Similarly, I2 bears theestimate

I2(t) ≤∥∥∥∂h∂t

∥∥∥L2#

′(Γ)

∥∥∥∂u∂t

∥∥∥L2#(Γ)

≤ 1

4ε

∥∥∥∂h∂t

∥∥∥2L2#

′(Γ)

+ ε∥∥∥∂u∂t

∥∥∥2L2#(Γ)

≤ 1

4ε

∥∥∥∂h∂t

∥∥∥2L2#

′(Γ)

+N22 ε∥∥∥∂u∂t

∥∥∥2L2(Ω)

+N22 ε∥∥∥∇∂u

∂t

∥∥∥2L2(Ω;Rn)

(8.179)

where N2 is the norm of the trace operator W 1,2(Ω) → L2#(Γ). Then we takeε > 0 small, namely (N2

1 +N22 )ε < 1, so that the last terms in (8.178)–(8.179) can

be absorbed in the corresponding term arising on the left-hand side of (8.175);note that (8.174) equals | ∂∂t∇u|2 if p = 2. Then we use Gronwall’s inequalityto handle the last-but-one terms in (8.178)–(8.179). Cf. also (8.70)–(8.71). Like

in (8.69a), it requires g ∈ W 1,2(I;L2∗′(Ω)) and h ∈ W 1,2(I;L2#

′(Γ)) only. Then

u ∈ W 1,2(I;W 1,2(Ω)).Assuming also u0 regular enough, namely u0 ∈ W 2,2(Ω) ∩ L2(q1−1)(Ω), and

g(0) ∈ L2(Ω), we have ∂∂tu(0) ∈ L2(Ω), and we can apply Gronwall’s inequality to

(8.175). We thus get the estimate for u in W 1,2(I;W 1,2(Ω)) ∩W 1,∞(I;L2(Ω)).(5) If both ∂

∂tu and f are functions (not only distributions), div(|∇u|p−2∇u)is more regular than W 1,p(Ω)∗, so that by elliptic regularity theory we obtain aspatial regularity. E.g., if p = 2, we can use the interior W 2,2- or W 3,2-regularityas established in Proposition 2.103; this needs 1 < q1 ≤ (2n−2)/(n−2) (andalso q1 ≥ 2 in the latter case). In combination with Steps (3) and (4), one thusobtains the last three lines in Table 3. If also Ω would be qualified, we could useProposition 2.104 to get regularity up to the boundary.

Exercise 8.64 (Large q1 or q2: an alternative setting). For q1 > p∗ or q2 > p#, wecan take the space W := {v ∈ Lp(I;W 1,p(Ω)) ∩ Lq1(Q); v|Σ ∈ Lq2(Σ)} endowednaturally by the norm ‖v‖W := ‖v‖Lp(I;W 1,p(Ω)) + ‖v‖Lq1(Q) + ‖v|Σ‖Lq2(Σ). Provethat W is a Banach space54, and that the monotone mapping A related to (8.167)maps W into W ∗ and that ∂

∂tu ∈ W ∗ if u is a weak solution to (8.167).55

54Hint: Consider a Cauchy sequence {uk}k∈N in W . Realize that, in particular, it convergesto u in Lp(I;W 1,p(Ω)) and in Lq1 (Q) as these spaces are complete, and also uk|Σ → u|Σ in

Lp(I;Lp# (Γ)), and, as {uk|Σ}k∈N is a Cauchy sequence in Lq2 (Σ), also uk|Σ → uΣ in thecomplete space Lq2 (Σ), and thus uΣ = u|Σ. Cf. also Exercise 2.72.

55Hint: Show A : W → W ∗ just by using Holder’s inequality. Further realize that ‖ ∂∂t

u‖W ∗ =

sup‖v‖W ≤1

∫Q |∇u|p−1∇u·∇v+ |u|q1−1uv−gv dx+

∫Σ |u|q2−1uv−hv dS and estimate it by the

Holder inequality.


qualification of quality

g h u0 p of u

Lp′(I;Lp∗′

(Ω)) Lp′(I;Lp#′

(Γ)) L2(Ω) >1 Lp(I;W 1,p(Ω))L∞(I;L2(Ω))

W 1,p′(I;W 1,p(Ω)∗)

W 1,1(I;Lp∗′(Ω)) W 1,1(I;Lp#′

(Γ)) W 1,p(Ω) >1 L∞(I;W 1,p(Ω))+ L2(Q) ∩L2(Ω) W 1,2(I;L2(Ω))

W 2,1(I;Lp∗′(Ω)) W 2,1(I;Lp#′

(Γ)) W 2,q(Ω) >1 W 1,∞(I;L2(Ω))+W 1,1(I;L2(Ω)) with L∞(I;W 1,p(Ω))

q≥ 2p∗

p∗−2p+4 ≥ 2 W 2/p−ε,p(I;W 1,p(Ω))

L2(Q) W 1,1(I;L2#′(Γ)) W 1,2(Ω) =2 L2(I;W 2,2

loc (Ω))

W 1,1(I;L2(Ω)) W 2,1(I;L2#′(Γ)) W 2,2(Ω) =2 L∞(I;W 2,2

loc (Ω))

W 1,2(I;L2∗′(Ω)) W 1,2(I;L2#

′(Γ)) W 2,2(Ω) =2 L2(I;W 3,2

loc (Ω))

∩L2(I;W 1,2loc (Ω))

Table 3. Summary of Example 8.63; qualification of q1 and q2 not displayed.

Example 8.65 (Nonmonotone term: a-priori estimates). Consider the initial-boundary-value problem with a nonmonotone term |u|μ instead of |u|q1−2u:

∂u

∂t− div

(|∇u|p−2∇u)+ |u|μ = g in Q,

|∇u|p−2 ∂u

∂ν+ |u|q2−2u = h on Σ,

u(0, ·) = u0 on Ω,

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (8.180)

where again g ∈ Lp′(I;Lp∗′

(Ω)) and h ∈ Lp′(I;Lp#′

(Γ)). We will show the a-prioriestimates again on an heuristical level only, and specify μ.

(1) The test by u(t, ·) itself now gives:

1

2

d

dt‖u‖2L2(Ω) + ‖∇u‖pLp(Ω;Rn) ≤

∫Ω

|u|μ+1 + gudx+

∫Γ

hu dS

≤ ‖u‖μ+1Lμ+1(Ω) +N

(‖g‖Lp∗′(Ω) + ‖h‖Lp#

′(Γ)

)‖u‖W 1,p(Ω) (8.181)

where N is as in (8.168). If μ ≤ 1, we can estimate

‖u‖μ+1Lμ+1(Ω) ≤ (measn(Ω))

(μ−1)/2‖u‖μ+1L2(Ω) ≤ (measn(Ω))

(μ−1)/2(‖u‖2L2(Ω) + c)

with some c > 0,56 and then use directly the Gronwall inequality. For superlinearlygrowing nonlinearities, i.e. μ > 1, ‖u‖μ+1

Lμ+1(Ω) can be absorbed in the left-hand side

56Cf. Exercise 2.58 for the norm of the embedding Lμ+1(Ω) ⊂ L2(Ω).


by using Young’s inequality through the estimate

‖u‖μ+1Lμ+1(Ω) ≤ Nμ+1‖u‖μ+1

W 1,p(Ω) ≤ Nμ+1(ε‖u‖pW 1,p(Ω) +Nε

)(8.182)

where N is the norm of W 1,p(Ω) ⊂ Lμ+1(Ω) and the last inequality uses

μ < p− 1; (8.183)

note that this implies also μ+ 1 ≤ p∗ used for the first inequality. This conditionmakes the approach effective only if p > 2 (otherwise the previous approach viathe Gronwall inequality can be used, too). The other terms can be estimatedin the same way as in (8.168). Again, this gives the a-priori estimate for u inLp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)).

(2) The estimate for ∂∂tu can be made just the same way as (8.169) or (8.172)

now with μ+1 in place of q1.(3) The test function v := ∂

∂tu(t, ·) needs again g ∈ L2(Q). For simplicity, wetake h = 0; otherwise, cf. Example 8.63(3). Then∥∥∥∂u

∂t

∥∥∥2L2(Ω)

+1

p

d

dt


≤∫Ω

|u|μ∣∣∣∂u∂t

∣∣∣ dx+

∫Ω

g∂u

∂tdx =: I1(t) + I2(t).

(8.184)

We can estimate I1(t) ≤ ε‖∂u∂t ‖2L2(Ω) + Cε‖u(t)‖2μL2μ(Ω). Assuming μ ≤ 1, we can

estimate simply∥∥u(t)∥∥2μL2μ(Ω)

≤ C(1 + ‖u(t)‖2L2(Ω)

) ≤ C

(1 + 2‖u0‖2L2(Ω) + 2T

∫ t

0

∥∥∥∂u∂t

∥∥∥2L2(Ω)

dt

),

cf. (8.63), and then use Gronwall inequality for (8.184). Thus we get the estimatefor u in L∞(I;W 1,p(Ω)) ∩W 1,2(I;L2(Ω)).

For 2 < p ≤ 2n/(n−2), we can afford a super-linear growth 1 < μ < p−1.Relying on (8.181) with (8.183), we have the estimate u ∈ L∞(I;L2(Ω)) at ourdisposal. Then we can use the interpolation of L2μ(Ω) between L2(Ω) andW 1,p(Ω),i.e., ∥∥u∥∥

L2μ(Ω)≤ C

∥∥u∥∥1−λ

L2(Ω)

∥∥u∥∥λW 1,p(Ω)

(8.185)

provided1

2μ≥ 1−λ

2+

λ(n−p)

np; (8.186)

cf. (1.39). Of course, we need 0 ≤ λ ≤ 1. Then, we can estimate ‖u(t, ·)‖2μL2μ(Ω) ≤C2μ‖u(t, ·)‖2λμW 1,p(Ω)‖u(t, ·)‖2(1−λ)μ

L2(Ω) and, assuming still

2λμ ≤ p, (8.187)

to treat it by the Gronwall inequality with help of the fact that ‖u(t, ·)‖L2(Ω) isa-priori bounded uniformly for t ∈ I. Thus we get again the estimate for u in


L∞(I;W 1,p(Ω)) ∩ W 1,2(I;L2(Ω)). It is important that, for p ≤ 2n/(n−2), theinequalities (8.186) and (8.187) are mutually consistent; we can just take λ =p/(2μ) which satisfies λ ≤ 1 if μ ≥ p/2, while for 1 < μ < p/2 we can firstestimate ‖u‖2μL2μ(Ω) ≤ C(1 + ‖u‖pL2μ(Ω)) and then follow as if μ = p/2.

For p > 2n/(n−2), we must slightly reduce the growth because the in-equalities (8.186) and (8.187) represent a certain restriction on μ, namely μ ≤np/(np− λ(np+2p−2n)) and μ ≤ p/(2λ), respectively. The optimal choice of λ isto make these bounds equal to each other, which gives λ = np/(np+2p) and thus

μ ≤ pn+2

2n. (8.188)

(4) Further, we apply ∂∂t to the equation and then use the test function

v = ∂∂tu. As in Example 8.63, we consider p ≥ 2 and now μ = 1; this is a model

case for arbitrary Lipschitz nonlinearities that could be treated by a modificationof this estimate. Then the term |u|μ = |u| can be estimated by∫

Ω

( ∂

∂t|u|)∂u∂t

dx =

∫Ω

dir(u)∣∣∣∂u∂t

∣∣∣2 dx ≤∥∥∥∂u∂t

∥∥∥2L2(Ω)

(8.189)

and then treated by the Gronwall inequality. The rest can be treated as in Exam-ple 8.63(4).

Exercise 8.66 (Limit passage). Suppose uk is the Galerkin solution for (8.180).Make the limit passage via Minty’s trick by using only the basic a-priori estimatesand monotonicity of the p-Laplacean.57 Alternatively, use d-monotonicity of thep-Laplacean and prove convergence directly without the Minty trick.58

Exercise 8.67 (Weaker estimate for ∂∂tu). Consider the problem (8.167) and derive

the estimate of ∂∂tu in M (I;W−k,2(Ω)) for k ∈ N so large that W k,2

0 (Ω) ⊂ L∞(Ω),i.e. k > p/n; this weaker estimate allows for bigger q1 and still suffices for usingAubin-Lions’ lemma as Corollary 7.9.59

Remark 8.68 (Regularized p-Laplacean). For p > 2, one can consider the parabolicproblem with a regularized p-Laplacean:

∂u

∂t− div

((ε+ |∇u|p−2

)∇u)= g, u(0) = u0, u|Σ = 0, (8.190)

cf. (4.38). This allows us to use the estimate from Step (4) from Example 8.63based on the uniform-like monotonicity (8.69c). Indeed, using also (8.174), wehave

∂

∂t

((ε+ |∇u|p−2)∇u

)·∂∇u

∂t≥ ε

∣∣∣∂∇u

∂t

∣∣∣2 + 4p−4

p2

( ∂

∂t|∇u|p/2

)2≥ ε

∣∣∣∂∇u

∂t

∣∣∣2.(8.191)

57Hint: Cf. Remark 8.32 modified by treating the non-monotone lower-order term |u|μ bycompactness as suggested in Exercise 8.59.

58Hint: Cf. Exercise 8.81 below.59Hint: Modify Example 8.63(2) by considering v ∈ C(I;W k,2

0 (Ω)) in (8.169).


Exercise 8.69. Consider again the regularized problem (8.190). Denoting uε itssolution, prove the a-priori estimates ‖uε‖Lp(I;W 1,p(Ω)) ≤ C, ‖uε‖L2(I;W 1,2(Ω)) ≤C/

√ε and ‖ ∂

∂tuε‖Lp′(I;W 1,p(Ω)∗) ≤ C/√ε, and then, passing ε → 0, prove uε → u

with u denoting the solution with ε = 0.60

Example 8.70 (Dirichlet boundary conditions). Let us illustrate the Dirichlet con-dition for a simple parabolic equation with the p-Laplacean, i.e.

∂u

∂t− div

(|∇u|p−2∇u)= 0, u(0, ·) = u0, u|Σ = uD |Σ, (8.192)

with some uD : Q → R prescribed. By multiplying the equation in (8.192) byv ∈ W 1,p

0 (Ω) and using Green’s formula, one gets the weak formulation:

∀(a.a.) t∈I ∀v∈W 1,p0 (Ω) :

⟨∂u∂t

, v⟩+

∫Ω

|∇u(t, x)|p−2∇u(t, x)·∇v(x) dx = 0

completed, of course, by u(0, ·) = u0 and u|Σ = uD|Σ.(1) We cannot test it by v = u(t, ·) if uD(t, ·)|Σ �= 0. Instead, the basic a-priori

estimate is obtained by a test by v = [u − uD](t, ·):61

1

2

d

dt

∥∥u−uD

∥∥2L2(Ω)

+∥∥∇u

∥∥pLp(Ω;Rn)

=

∫Ω

|∇u|p−2∇u·∇uD − ∂uD

∂t(u−uD) dx

≤ ε∥∥∇u

∥∥pLp(Ω;Rn)

+ Cε

∥∥∇uD

∥∥pLp(Ω;Rn)

+∥∥∥∂uD

∂t

∥∥∥L2(Ω)

(1 +

∥∥u−uD

∥∥2L2(Ω)

).

(8.193)

If uD ∈ W 1,p,1(I;W 1,p(Ω), L2(Ω)) and u0 ∈ L2(Ω), by Gronwall’s inequality weobtain u−uD bounded in L∞(I;L2(Ω)). As uD ∈ L∞(I;L2(Ω)) due to Lemma 7.1,also u itself bounded in L∞(I;L2(Ω)). Integrating still (8.193) over [0, T ] we get ubounded in Lp(I;W 1,p(Ω)). Then, from the equation (8.192) itself, one gets ∂

∂tu

bounded in Lp′(I;W 1,p

0 (Ω)∗).

60Hint: Use Minty’s trick:

0 ≤∫Q

(|∇uε|p−2∇uε − |∇v|p−2∇v + ε∇(uε−v))·∇(uε−v) dxdt

=

∫Q

(g − ∂uε

∂t

)(uε−v) − |∇v|p−2∇v·∇(uε−v) − ε∇v·∇(uε−v) dxdt

and realize that∫Q ε∇v · ∇(uε−v) dxdt = O(

√ε). Then put v = u+ δw, and pass δ > 0 to zero.

61Equally, one can apply the shift (2.61), i.e. hereA0(t, u) = A(u+uD(t)). Then, writing u(t) :=u−uD(t), the original equation d

dtu+A(u) = f is equivalent to d

dtu+A0(t, u) = f0:=f− d

dtuD and

its test by v := u(t, ·) ∈W 1,p0 (Ω) is precisely (8.193) in the special case f = 0.


(2) Further estimates can be obtained by testing by v = ∂∂t [u− uD](t, ·):∥∥∥∂u

∂t

∥∥∥2L2(Ω)

+1

p

d

dt


=

∫Ω

|∇u|p−2∇u·∇∂uD

∂t+

∂u

∂t

∂uD

∂t

≤ ∥∥ |∇u|p−1∥∥Lp′(Ω)

∥∥∥ ∣∣∇∂uD

∂t

∣∣ ∥∥∥Lp(Ω)

+∥∥∥∂u∂t

∥∥∥L2(Ω)

∥∥∥∂uD

∂t

∥∥∥L2(Ω)

≤(1+∥∥∇u

∥∥pLp(Ω;Rn)

)∥∥∥∇∂uD

∂t

∥∥∥Lp(Ω;Rn)

+1

4

∥∥∥∂u∂t

∥∥∥2L2(Ω)

+∥∥∥∂uD

∂t

∥∥∥2L2(Ω)

. (8.194)

Here one needs ∂∂tuD ∈ L2(Q)∩L1(I;W 1,p(Ω)) and u0 ∈ W 1,p(Ω) to get u bounded

in L∞(I;W 1,p(Ω)) ∩W 1,2(I;L2(Ω)).62

(3) Still further estimates can be obtained by differentiating (8.192) in timeand by testing it by v = ∂

∂t [u− uD](t, ·). In view of (8.174), the term thus arisingon the right-hand side can be estimated as∣∣∣∣ ∂∂t(|∇u|p−2∇u

)·∇∂uD

∂t

∣∣∣∣ = ∣∣∣∣ |∇u|p−2 ∂∇u

∂t· ∂∇uD

∂t

+((p−2)|∇u|p−4∇u·∂∇u

∂t

)(∇u·∂∇u

∂t

)∣∣∣∣≤ (p−1)|∇u|p−2

∣∣∣∂∇u

∂t

∣∣∣ ∣∣∣∂∇uD

∂t

∣∣∣≤ p−1

2|∇u|p−2

∣∣∣∂∇u

∂t

∣∣∣2+ p−1

2|∇u|p−2

∣∣∣∂∇uD

∂t

∣∣∣2=: I1 + I2. (8.195)

The term I1 can be absorbed in term arising from (8.174) on the left-hand sidewhich just equals 1

2I1, while, assuming uD ∈ W 1,p(I;W 1,p(Ω)), the term I2 bearsthe estimate ∫

Q

I2 dxdt ≤ p−1

2

∥∥∇u∥∥p−2

Lp(Q;Rn)

∥∥∥∂∇uD

∂t

∥∥∥2Lp(Q;Rn)

. (8.196)

Altogether, like (8.175), one gets

1

2

d

dt

∥∥∥∂u∂t

∥∥∥2L2(Ω)

+2p−2

p2

∥∥∥∂|∇u|p/2∂t

∥∥∥2L2(Ω)

≤ p−1

2

∥∥∇u∥∥p−2

Lp(Q;Rn)

∥∥∥∂∇uD

∂t

∥∥∥2Lp(Q;Rn)

.

Integrating it on [0, t] for a general t ∈ I, one obtains ∂∂tu ∈ L∞(I;L2(Ω)) and

∂∂t |∇u|p/2 ∈L2(Q) if also ∂

∂tu(0)= div(|∇u0|p−2∇u0)∈L2(Ω); this last qualifica-tion is satisfied if u0 ∈ W 2,q(Ω) with q ≥ 2p∗/(p∗−2p+4) as in Example 8.63(4).

62In view of Theorem 8.16(iii), one may think that u(t)−uD(t) ∈ W 1,p0 (Ω) so that, in particular,

u0|Γ = uD on Γ. Yet, here we should rather consider the W 1,1-structure of the loading uD (likein Remark 8.23) to be extended for a BV-loading that does not need to be continuous at t = 0so that even u0|Γ �= uD on Γ gives the expected a-priori estimate.


Note that, in particular, uD and u0 qualify for using Step (1), hence certainly∇u ∈ Lp(Q;Rn), so that the right-hand side of (8.196) is indeed finite. As inExample 8.63(4), we also get ∇u ∈ W 2/p−ε,p(I;Lp(Ω;Rn)) if p ≥ 2.

qualification of qualityuD u0 of u

W 1,p,1(I;W 1,p(Ω), L2(Ω)) L2(Ω) W 1,p,p′(I;W 1,p(Ω),W 1,p(Ω)∗)L∞(I;L2(Ω))

W 1,2(I;L2(Ω)) ∩W 1,1(I;W 1,p(Ω)) W 1,p(Ω) L∞(I;W 1,p(Ω))W 1,2(I;L2(Ω))

W 1,p(I;W 1,p(Ω)) W 2,q(Ω) W 1,∞(I;L2(Ω))

q≥ 2p∗p∗−2p+4 p≥2 : W 2/p−ε,p(I;W 1,p(Ω))

Table 4. Summary of Example 8.70.

8.8.3 Semilinear heat equation c(u) ∂∂tu− div(κ(u)∇u) = g

In this subsection, we will scrutinize the heat transfer in nonlinear but homoge-neous isotropic media, described by the heat equation for the unknown θ (insteadof u) which has the interpretation as temperature:

c(θ)∂θ

∂t− div(κ(θ)∇θ) = g (8.197)

where

θ : Q → R is the unknown temperature,κ : R → R+ is the heat conductivity,

c : R → R+ is the heat capacity,g the volume heat sources,

cf. Example 2.71 for a steady-state variant.

Example 8.71 (Enthalpy transformation). Powerful tools for nonlinear differentialequations are various transformations of independent variables. Here we can apply,besides the Kirchhoff transformation, also the enthalpy transformation:

c (r) :=

∫ r

0

c(�) d� & κ (r) :=

∫ r

0

κ(�) d� . (8.198)

Putting u := c (θ) (u is called enthalpy) and denoting β(u) := [ κ ◦ c−1](u), wehave ∂

∂tu = [ c ]′(θ) ∂∂tθ = c(θ) ∂

∂tθ and Δ(β(u)) = div( κ ′( c−1(u))∇ c−1(u)) =div( κ ′(θ)∇θ) = div(κ(θ)∇θ). This transforms the original equation (8.197) to∂∂tu−Δβ(u) = g. Considering the initial condition in terms of the enthalpy u0 and

the boundary condition as in Example 2.71, i.e. κ(θ) ∂∂ν θ = b1(θe−θ)+b2(θe−|θ|3θ)

with θe an external temperature, we come to the following initial-boundary-value


problem:

∂u

∂t−Δβ(u) = g in Q,

∂β(u)

∂ν+(b1 + b2| κ−1(u)|3

)κ−1(u) = h on Σ,

u(0, ·) = u0 on Ω.

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(8.199)

Exercise 8.72 (Pseudomonotone approach). The nonlinear operator −Δβ(u) =−div

(β′(u)∇u

)can be considered as pseudomonotone and treated by Proposi-

tions 8.38 and 8.40. Assuming β∈C1(R), verify (8.141), (8.142), and (8.143) in thisspecial case.63 Realize, in particular, the condition 0 < inf β′(R) ≤ supβ′(R) <+∞.

Exercise 8.73 (Weak-continuity approach). Realizing that the operator−div

(β′(u)∇u

)is semilinear in the sense (8.161), one can use Proposition 8.47.

Assume, besides β∈C1(R), the growth restriction

0 < ε ≤ β′(r) ≤ C(1 + |r|(2�−ε)/2

)(8.200)

for some ε > 0 and 2� = n+ 4/n, cf. (8.131). Verify (8.12) for q = ∞ andZ = W 1,∞(Ω), and also (8.161), (8.162) and (8.163).64

Assuming g ∈ L2(I;L2∗′(Ω)) + L1(I;L2(Ω)), h ∈ L2(I;L2#

′(Γ)) and u0 ∈

L2(Ω), prove the a-priori estimates of u in L2(I;W 1,2(Ω))∩L∞(I;L2(Ω)), and of∂∂tu in L1(I;W 1,∞(Ω)∗), and the convergence of approximate solutions to a veryweak solution.65 Realize, in particular, that now the heat conductivity κ(·) neednot be bounded, e.g. if n = 3, then κ(r) = 1 + |r|q1 with q1 < 5/3 is admittedif c(·) ≥ ε > 0. Also note that ∂

∂tu, living in L1(I;W 1,∞(Ω)∗) in general, is notin duality with u, hence the concept of the very weak solution is indeed essential

63Hint: Realize that here a(t, x, r, s) = β′(r)s, b(t, x, r) = (b1(t, x)+b2(t, x)| κ−1(r)|3)κ−1(r),

and c(t, x, r, s) = 0. Then (8.141) needs β′ > 0, but (8.142)–(8.143) needs p = 2 and β′ boundedand away from zero.

64Hint: Obviously a(t, x, r, s) = β′(r)s is of the form (8.161a) and then realize that the upperbound in (8.200) is just (8.162a) while (8.163) needs just the lower bound in (8.200). As to (8.12),

use (8.200) and the interpolation between L2(Ω) and L2∗ (Ω) with λ from (8.145) to estimate

sup‖v‖

W1,∞(Ω)≤1

∫Ωβ′(u)∇u · ∇v dx ≤ ‖β′(u)‖L2(Ω)‖∇u‖L2(Ω;Rn)

≤ ∥∥1+|u|2�/2∥∥L2(Ω)

‖∇u‖L2(Ω;Rn) ≤ C(measn(Ω)1/2+ ‖u‖2�/2

L2� (Ω)

)‖∇u‖L2(Ω;Rn)

≤ C(measn(Ω)1/2+ ‖u‖λ2�/2

L2(Ω)‖u‖(1−λ)2p

∗/2

L2∗ (Ω)

)‖∇u‖L2(Ω;Rn).

65Hint: Considering e.g. Galerkin approximate solutions uk, by Aubin-Lions Lemma 7.7, uk →u holds in L2�−ε(Q). By (8.200), we have β′(uk)→ β′(u) in L2(Q). Then make the limit passage∫Ω∇β(uk) · ∇v dx =

∫Ω β′(uk)∇uk · ∇v dx→ ∫

Ω β′(u)∇u · ∇v dx =∫Ω∇β(u) · ∇v dx.


if β′(·) is not bounded. Another occurrence of this effect is under an advectiondriven by a velocity field which is not regular enough, see Lemma 12.14.

Exercise 8.74 (Semi-implicit time discretization). Consider the linearization ofthe nonlinear “heat-transfer” operator related to (8.199), namely

[B(w, u)

](v) :=∫

Ωβ′(w)∇u · ∇v dx +

∫Γ

(b1u + b2|w|3u

)v dS, and then the semi-implicit formula

(8.55), which leads to⟨∂uτ

∂t, v⟩+

∫Ω

β′(uR

τ )∇uτ ·∇v − gτv dx =

∫Γ

(b1uτ + b2|uR

τ |3uτ

)v dS (8.201)

for a.a. t∈I and all v=W 1,2(Ω) with the ‘retarded’ Rothe function uRτ defined by

uR

τ (t, ·) :={

uτ (t− τ, ·) for t ∈ [τ, T ],uτ (0, ·) for t ∈ [0, τ ].

(8.202)

Assuming (8.200), make a basic a-priori estimate by a test by ukτ and prove the

convergence for τ → 0.66

Example 8.75 (Heat equation with advection). The heat transfer in a mediummoving with a prescribed velocity field �v : Q → Rn is governed by the equation

c(θ)(∂θ∂t

+ �v · ∇θ)− div

(κ(θ)∇θ

)= g. (8.203)

In the enthalpy formulation from Example 8.71 it reads as ∂∂tu+�v·∇u+Δβ(u) = g.

Assuming div�v ≤ 0 and (�v|Σ) · ν ≥ 0 as in Exercise 2.91 and using (6.33), themapping A(t, u) : W 1,2(Ω) → W 1,∞(Ω)∗ defined by 〈A(t, u), z〉 :=

∫Ωβ′(u)∇u ·

∇z + (�v(t, ·) · ∇u) z dx can be shown semi-coercive if β′ satisfies (8.200):⟨A(t, u), u

⟩=

∫Ω

β′(u)|∇u|2 + (�v(t, ·) · ∇u)u dx ≥ ε‖∇u‖2L2(Ω;Rn). (8.204)

In case div�v = 0 and (�v|Σ) ·ν = 0, the scalar variant of (6.35) yields∫Ω(�v(t, ·) ·

∇u) z dx = − ∫Ω(�v(t, ·) · ∇z)u dx ≤ ‖�v(t, ·)‖L2(Ω;Rn)‖∇z‖L∞(Ω;Rn)‖u‖L2(Ω) and,expanding Exercise 8.73, we can rely on the concept of a very weak solution

u ∈ W 1,2,1(I;W 1,2(Ω),W 1,∞(Ω)∗) if �v ∈ L1(I;W1,2n/(n+2)0,div (Ω;Rn)). If β′(·) is

bounded and �v∈L∞(I;L2∗2/(2∗2−2∗−2)(Ω)), by the estimate∫Ω(�v(t, ·)·∇u) z dx ≤

‖�v(t, ·)‖L2∗2/(2∗2−2∗−2)(Ω)‖∇u‖L2(Ω;Rn)‖z‖L2∗(Ω), we can rely on the conventional

concept of a weak solution u ∈ W 1,2,2(I;W 1,2(Ω),W 1,2(Ω)∗) as in Exercise 8.72.

8.8.4 Navier-Stokes equation ∂∂tu+(u·∇)u−Δu+∇π=g, div u=0

Another important example of a semilinear equation, or rather a system of equa-tions, is the evolution version of the Navier-Stokes equation, cf. Remark 6.15,

∂u

∂t+ (u·∇)u−Δu +∇π = g, div u = 0, u(0, ·) = u0, (8.205)

66Hint: Cf. Exercise 8.92 below.


for the velocity field u : Q → Rn and the pressure π : Q → R which is, in fact, amultiplicator to the constraint div u = 0. This system is a model for a flow of aviscous incompressible fluid; the viscosity and the mass density is put equal to 1 in(8.205). Considering zero Dirichlet boundary conditions, we pose the problem intofunction spaces by putting V := W 1,2

0,div(Ω;Rn) = {v ∈W 1,2

0 (Ω;Rn); div u = 0},cf. (6.29), endowed with the norm ‖v‖V := ‖∇v‖L2(Ω;Rn×n) and, to ensure V ⊂ Hdensely, we defineH as a closure of V in L2(Ω;Rn) in the L2-norm. Then, in accordwith Definition 8.1 (and Table 2. on p.215), u ∈ W 1,2,2(I;V, V ∗) is considered asa weak solution to (8.205) if u(0, ·) = u0 and if⟨∂u

∂t, v⟩V ∗×V

+

∫Ω

(u·∇)u · v +∇u : ∇v − gv dx = 0 (8.206)

for any v ∈ V and for a.a. t ∈ I. Naturally, here the mapping A : V → V ∗ isdefined by 〈A(u), v〉 :=

∫Ω(u·∇)u · v + ∇u : ∇v dx; this definition is correct for

n ≤ 3.67 Coercivity of A can be obtained by using (6.36): indeed it holds simplythat 〈A(v), v〉 = ∫Ω∇v : ∇v + ((v · ∇) v) · v dx = ‖∇v‖2L2(Ω;Rn×n).

Example 8.76 (Pseudomonotone approach). For n = 2, we can estimate

sup‖v‖V ≤1

∣∣∣ ∫Ω

((u ·∇)u

) · v dx∣∣∣ = sup‖v‖V ≤1

∫Ω

((u ·∇) v

) · u dx≤ sup‖v‖V ≤1

∥∥u∥∥2L4(Ω;Rn)

∥∥∇v∥∥L2(Ω;Rn×n)

≤ C2GN

∥∥u∥∥L2(Ω;Rn)

∥∥∇u∥∥L2(Ω;Rn×n)

(8.207)

where the Holder inequality and the Gagliardo-Nirenberg inequality like (1.40)has been used. Hence A satisfies the growth condition (8.13) with p = q = 2 andC(ζ) = max(1, ζ). Hence we have guaranteed existence of a weak solution if u0 ∈ Hand g ∈ L2(I;L2∗′

(Ω;Rn)).

Example 8.77 (Weak-continuity approach). We consider the physically relevantcase n = 3 (which covers n = 2 too), and will verify the condition (8.13) withZ = V . We have the embedding W 1,2(Ω) ⊂ L6(Ω); let us denote by N its norm,and estimate by the Holder inequality and an interpolation:

‖A(u)‖V ∗ ≤ sup‖∇v‖L2(Ω;Rn×n)≤1

∫Ω

∇u · ∇v + (u · ∇u)v dx

≤ sup‖∇v‖L2(Ω;Rn×n)≤1

‖∇u‖L2(Ω;Rn×n)‖∇v‖L2(Ω;Rn×n)

+ ‖u‖L3(Ω;Rn)‖∇u‖L2(Ω;Rn×n)‖v‖L6(Ω;Rn)

67Since 4 > 2∗ for n ≤ 3, this follows from the Holder inequality | ∫Ω(u·∇)u · v dx| ≤‖u‖L4(Ω;Rn)‖∇u‖L2(Ω;Rn×n)‖v‖L4(Ω;Rn). In fact, using Gagliardo-Nirenberg inequality, the bor-derline case n = 4 can be covered, too.


≤ ‖∇u‖L2(Ω;Rn×n) +N‖u‖1/2L6(Ω;Rn)‖u‖1/2L2(Ω;Rn)‖∇u‖L2(Ω;Rn×n)

≤ ‖u‖V +N3/2‖u‖1/2H ‖u‖3/2V ≤ max(1, N3/2‖u‖1/2H )(1+‖u‖3/2V

).

Hencefore, we get (8.13) with p = 2, q = 4 and C(r) = max(1, N3/2r1/2). As nowq′ = 4/3, the estimate (8.19a) yields ∂

∂tu∈L4/3(I;V ∗), which, however, is not induality with L2(I;V ) � u and the concept of the very weak solution and weakcontinuity is indeed urgent.68

Remark 8.78 (Uniqueness). We consider n = 2. Taking two weak solutions u1 andu2, subtracting the respective identities (8.206), testing it by v = u12 := u1 −u2, and using subsequently (6.36), the Holder inequality, the Gagliardo-Nirenberginequality like (1.40), one obtains

1

2

d

dt‖u12‖2L2(Ω;Rn)+ ‖∇u12‖2L2(Ω;Rn×n) =

∫Ω

((u1·∇)u1 − (u2·∇)u2

)·u12dx

=

∫Ω

(u12·∇)u2 · u12 + (u2·∇)u12 · u12 dx

=

∫Ω

(u12·∇)u2 · u12 dx ≤ ‖∇u2‖L2(Ω;Rn×n)‖u12‖2L4(Ω;Rn)

≤ C2GN

‖∇u2‖L2(Ω;Rn×n)‖u12‖L2(Ω;Rn)‖∇u12‖L2(Ω;Rn×n)

≤ 1

2C4

GN‖∇u2‖2L2(Ω;Rn×n)‖u12‖2L2(Ω;Rn) +

1

2‖∇u12‖2L2(Ω;Rn×n)

with CGN

the constant from the Gagliardo-Nirenberg inequality ‖v‖L4(Ω) ≤CGN‖v‖1/2L2(Ω)‖∇v‖1/2L2(Ω;Rn). Then absorbing the last term in the left-hand side and

using the Gronwall inequality when realizing that t �→ ‖∇u2(t, ·)‖2L2(Ω;Rn×n) ∈L1(I) and u12(0, ·) = 0, one obtains u12(t, ·) = 0 for a.a. t ∈ I.

This technique does not work if n = 3 and surprisingly, in this physicallyrelevant case, the uniqueness is a mysteriously difficult problem.69

Exercise 8.79. Derive the weak formulation (8.206).70

Exercise 8.80 (Darcy-Brinkman system). Modify the analysis of (8.205) in thissection by adding another, lower-order dissipative term, namely

∂u

∂t+ (u·∇)u−Δu+ u+∇π = g, div u = 0, u(0, ·) = u0; (8.208)

68Note that the idea of putting Z = V ∩W 1,∞(Ω;Rn) would lead to ∂∂t

u ∈ L2(I;Z∗) which

is again not in duality with L2(I; V ).69This question, intimately related to regularity for (8.205), was identified by the Clay Math-

ematical Institute as one out of 7 most challenging mathematical “Millennium problems”, andat the time of publishing even the 2nd edition of this book was still waiting for its (affirmativeor not) answer, together with a $ 1 million award.

70Hint: Test (8.205) by v ∈ V , integrate it over Ω, and use Green’s formula and the orthogo-nality

∫Ω(∇π)v dx = − ∫

Ω π div v dx = 0.


this additional term bears an interpretation as a reaction force to the flow throughstatic porous media with a constant porosity.

8.8.5 Some more exercises

Exercise 8.81. Consider the parabolic problem:

∂u

∂t− div

(|∇u|p−2∇u)+ c(∇u) = g, u(0) = u0, u|Σ = 0, (8.209)

with the continuous nonlinearity c : Rn → R having the growth restricted by

|c(s)| ≤ C(1 + |s|p−1−δ

)(8.210)

with some δ > 0, and show the basic a-priori estimates of an approximate solutionobtained by the Galerkin method if u0 ∈ L2(Ω) and g ∈ Lp′

(I;Lp∗′(Ω)).71 Show the

existence of a weak solution to (8.209) by convergence of Galerkin’s solutions uk byusing the d-monotonicity of the p-Laplacean to prove first the strong convergenceof ∇uk, similarly as in Exercise 2.85.72

71Hint: Test the identity∫Ω

(∂∂t

uk+c(∇uk)−g)v + |∇uk|p−2∇uk·∇vdx = 0 by v:=uk(t, ·):

1

2

d

dt

∥∥uk

∥∥2L2(Ω)

+∥∥∇uk

∥∥pLp(Ω)

=

∫Ω

(g − c(∇uk)

)ukdx

≤ ε∥∥u∥∥p

Lp∗(Ω)+ 2p

′−1Cε

(∥∥c(∇uk)∥∥p′

Lp∗′(Ω)

+∥∥g∥∥p′

Lp∗′(Ω)

),

from which the a-priori estimate of uk in L∞(I;L2(Ω)) ∩ Lp(I;W 1,p(Ω)) follows by Gronwall’sinequality and by using (8.210), so that

∥∥c(∇uk)∥∥p′

Lp∗′(Ω)

≤∫ΩCp′(1 + |∇uk|p−1−δ

)p′dx ≤ Cε,δ + ε

∥∥∇uk

∥∥pLp(Ω;Rn)

.

The dual estimate of ∂∂t

uk in Lp′(I;W 1,p0 (Ω)∗lcs) can then be obtained standardly.

72Hint: Take a subsequence uk ⇀ u in W 1,p,p′(I;W 1,p0 (Ω),W 1,p

0 (Ω)∗lcs). Use the norm‖v‖

W1,p0 (Ω)

:= ‖∇v‖Lp(Ω;Rn) and, by (2.141), estimate

(‖uk‖p−1

Lp(I;W1,p0 (Ω))

− ‖v‖p−1

Lp(I;W1,p0 (Ω))

)(‖uk‖Lp(I;W

1,p0 (Ω))

− ‖v‖Lp(I;W

1,p0 (Ω))

)≤

∫Q

(|∇uk|p−2∇uk − |∇v|p−2∇v)(∇uk −∇v

)dxdt

=

∫Q|∇uk|p−2∇uk·

(∇uk−∇vk)+ |∇uk|p−2∇uk·

(∇vk−∇v)− |∇v|p−2∇v·(∇uk−∇v

)dxdt

=

∫Q

(g−c(∇uk)−

∂uk

∂t

)(uk−vk) + |∇uk|p−2∇uk·

(∇vk−∇v)− |∇v|p−2∇v·(∇uk−∇v

)dxdt

with vk(t, ·) ∈ Vk. Assume vk → v in Lp(I;W 1,p(Ω))). For v = u, uk− vk → u−u = 0 in Lp(Q)

because of the compact embedding W 1,p0 (Ω) � Lp(Ω) and Aubin-Lions’ Lemma 7.7, and then∫

Ωc(∇uk)(uk − vk)dx→ 0 because {c(∇uk)}k∈N is bounded in Lp′(Q) thanks to (8.210). Use

lim supk→∞

∫Q−∂uk

∂tukdxdt = lim sup

k→∞

∫Ω

u20k − uk(T )2

2dx ≤

∫Ω

u20 − u(T )2

2dx =

∫ T

0−⟨∂u

∂t, u

⟩dt


Exercise 8.82. Consider again the parabolic problem (8.209) with c satisfying

|c(s)| ≤ C(1 + |s|p/2) (8.211)

and show existence of a weak solution to (8.209) if p > 2n/(n+2), u0 ∈W 1,p

0 (Ω) and g ∈ L2(Q) in a simpler way than in Exercise 8.81 by using theL2(Q)-estimate on ∂

∂tu and convergence of Galerkin’s approximations weakly in

W 1,∞,2(I;W 1,p0 (Ω), L2(Ω)) and strongly in Lp(I;W 1,p

0 (Ω)).73

Exercise 8.83. Show how the coercivity works in the above concrete cases. Checkthe coercivity (8.95) or (8.60d).74

Exercise 8.84. Prove a-priori estimates and convergence of Galerkin approximantsfor the equation ∂

∂tu−div(|∇u|p−2∇u+|u|μ∇u) = g, with p > 2 and some μ ≥ 0.75

Exercise 8.85 (Singular perturbations by a biharmonic-term). Consider

∂u

∂t− div(|∇u|p−2∇u) + εΔ2u = g, u(0, ·) = u0, u|Σ =

∂u

∂ν

∣∣∣Σ= 0. (8.212)

Denoting uε the weak solution to (8.212), execute the test by uε and prove a-priori

because uk(T ) ⇀ u(T ) weakly in L2(Ω), cf. (8.104). Push the other terms to zero, too. Conclude

that uk → u in Lp(I;W 1,p0 (Ω)). Finally, pass to the limit directly in the Galerkin identity, which

gives the integral identity∫ T0

(〈 ∂∂t

u, v〉+ ∫Ω |∇u|p−2∇u·∇v+ c(∇u)v − gv dx

)dt = 0. Choosing

v(t, x) = ζ(t)z(x) with ζ ∈ C0(I) and z ranging a dense subset of W 1,p(Ω), prove the pointwise(for.a.a. t) equation of the type (8.126).

73Hint: A further test of the Galerkin identity by ∂∂t

uk gives

∥∥∥∂uk

∂t

∥∥∥2L2(Ω)

+1

p

d

dt

∥∥∇uk

∥∥pLp(Ω)

=

∫Ω

(g−c(∇uk)

)∂uk

∂tdx ≤ ∥∥c(∇uk)− g

∥∥2L2(Ω)

+∥∥∥∂uk

∂t

∥∥∥2L2(Ω)

,

from which the a-priori estimate of uk in W 1,2(I;L2(Ω)) ∩ L∞(I;W 1,p(Ω))) follows by Gron-wall’s inequality by using (8.211), so that ‖c(∇uk)‖2L2(Ω)

≤ 2C2(measn(Ω) + ‖∇uk‖pLp(Ω)).

The convergence of uk to some u solving (8.209) can be made similarly as in Exercise 8.81but we can make directly the limit passage limk→∞

∫Q( ∂

∂tuk)ukdxdt =

∫Q( ∂

∂tu)udxdt because

∂∂t

uk ⇀ ∂∂t

u weakly in L2(Q) and uk ⇀ u weakly* in W 1,∞,2(I;W 1,p0 (Ω), L2(Ω)), hence by

Aubin-Lions’ Lemma 7.7 strongly in L2(Q) provided p > 2n/(n+2) so that W 1,p0 (Ω) � L2(Ω).

74Hint: Note that, e.g. for the case (8.180),

⟨A(v), v

⟩=

∫Ω|∇v|p + |v|q1 + |v|μv dx+ b

∫Γ|v|q2dS ≥ c

∥∥v∥∥q

W1,p(Ω)− ∥∥v∥∥μ+1

Lμ+1(Ω)− C

where we used an equivalent norm on W 1,p(Ω). In particular, the semi-coercivity (8.95) holds

for μ ≤ 1 or p > μ+1, q2 ≥ p (but q2 ≤ p∗′), q1 ≥ 1 (but q1−1 ≤ p#′), and b > 0. Alternatively,

q2 ≥ 1 is sufficient if q1 ≥ p. The weaker coercivity (8.60d) holds even for q2 > 1 and q1 ≥ 1 orvice versa q2 ≥ 1 and q1 > 1.

75Hint: Use the test by uk to get {uk}k∈N bounded in Lp(I;W 1,p(Ω)) ∩ L∞(I;L2(Ω)),then estimate ∂

∂tuk, and show convergence, e.g., by Minty’s trick for Δp combined with

limsupk→∞∫Q∇uk · ∇(uk − v) dxdt ≤ ∫

Q∇u · ∇(u− v) dxdt.


estimates:∥∥uε

∥∥L2(I;W 2,2

0 (Ω))≤ C/

√ε,

∥∥uε

∥∥Lp(I;W 1,p

0 (Ω))∩L∞(I;L2(Ω))≤ C, (8.213a)∥∥∥∂uε

∂t

∥∥∥L1(I;W−2,2(Ω)+W−1,p′(Ω))

≤ C,∥∥∥∂uε

∂t+εΔ2uε

∥∥∥Lp′(I;W−1,p′(Ω))

≤ C, (8.213b)

assuming that g ∈ Lp′(I;Lp∗′

(Ω)) and u0 ∈ L2(Ω). Selecting a subsequence uε → uconverging weakly* in Lp(I;W 1,p

0 (Ω))∩L∞(I;L2(Ω)) for ε → 0, show the conver-gence76

∂uε

∂t+εΔ2uε → ∂u

∂tweakly in Lp′

(I;W−1,p′(Ω)). (8.214)

Also show that77

uε(T ) → u(T ) weakly in L2(Ω). (8.215)

Then show the convergence of uε to the weak solution of the initial-boundary-value problem ∂

∂tu − div(|∇u|p−2∇u) = g, u(0, ·) = u0, and u|Σ = 0.78 Assuming

g ∈ L2(Q) and u0 ∈ W 1,p0 (Ω), and regularizing the initial condition uε(0) = u0ε ∈

W 2,20 (Ω) ∩W 1,p(Ω) so that u0ε → u0 in W 1,p

0 (Ω) while ‖u0ε‖W 2,2(Ω) = O(1/√ε),

execute the test by ∂uε

∂t to prove the a-priori estimates:

∥∥uε

∥∥L∞(I;W 1,p

0 (Ω))≤ C,

∥∥uε

∥∥L∞(I;W 2,2

0 (Ω))≤ C√

ε,∥∥∥∂uε

∂t

∥∥∥L2(Q)

≤ C. (8.216)

Modify this example by considering other boundary conditions or also some otherterms as div(a(u)) or a quasilinear regularizing term as in Example 2.46.

Remark 8.86 (Strong convergence of the singular perturbations). Strong conver-gence in Example 8.85 based on (8.213) is expectable but rather technical, however.

We will use∫ T

0〈∂uε

∂t − ∂u∂t , uε−u〉dt ≥ 0; here again it is important that ∂u

∂t belongs

76Hint: By (8.213b), choose ∂uε∂t

+εΔ2uε → u ∈ Lp′(I;W−1,p′(Ω)) and, for any z ∈ D(Q), use

⟨u, z

⟩= lim

ε→0

⟨∂uε

∂t+ εΔ2uε, z

⟩= lim

ε→0

∫Q

∂uε

∂tz + ε∇2vε:∇2z dxdt = − lim

ε→0

∫Quε

∂z

∂t− ε∇2uε:∇2z dxdt = −

∫Qu∂z

∂tdxdt

because, thanks to (8.213a), ‖ε∇2uε‖L2(Q;Rn×n) = εO(1/√ε) = O(

√ε) → 0. Thus u is shown

to be the distributional derivative of u.77Hint: Use (8.214) and the first estimate in (8.213a) to show the W−1,p′ -weak convergence:

uε(T ) = u0 +

∫ T

0

∂uε

∂tdt = v0 +

∫ T

0

∂uε

∂t+εΔ2uε dt− ε

∫ T

0Δ2uε dt→ u0 +

∫ T

0

∂u

∂tdt = u(T )

and then the second estimate in (8.213a) implies (8.215).78Hint: Use Minty’s trick based on the monotonicity of the operator ∂

∂t−Δp + εΔ2 similarly

like in Exercise 2.100.


to Lp′(I;W 1,p

0 (Ω)∗) and is thus in duality to u. By the d-monotonicity (2.141) of−Δp, we get, for any u ∈ L2(I;W 2,2

0 (Ω)) ∩ Lp(I;W 1,p(Ω)):(∥∥∇uε

∥∥p−1

Lp(Q;Rn)− ∥∥∇u

∥∥p−1

Lp(Q;Rn)

)(∥∥∇uε

∥∥Lp(Q;Rn)

− ∥∥∇u∥∥Lp(Q;Rn)

)≤∫Q

(|∇uε|p−2∇uε − |∇u|p−2∇u)·∇(uε − u) dxdt

≤∫ T

0

(⟨∂uε

∂t− ∂u

∂t, uε−u

⟩+

∫Ω

(|∇uε|p−2∇uε− |∇u|p−2∇u)·∇(uε−u) dx

)dt

=

∫Q

(∂uε

∂t· uε + |∇uε|p−2∇uε·∇uε

)−(∂uε

∂t· u+ |∇uε|p−2∇uε·∇u

)dxdt

−∫ T

0

(⟨∂u∂t

, uε−u⟩+

∫Ω

|∇u|p−2∇u·∇(uε−u) dx

)dt

=

∫Q

(− ε|∇2uε|2 + guε + ε∇2uε :∇2u− gu

− ∂uε

∂t(u−u)− |∇uε|p−2∇uε·∇(u−u)

)dxdt

−∫ T

0

(⟨∂u∂t

, uε−u⟩+

∫Ω


)dt

≤∫Q

ε∇2uε:∇2u+ g(uε−u)− ∂uε

∂t(u−u)− |∇uε|p−2∇uε·∇(u−u) dxdt

−∫ T

0

(⟨∂u∂t

, uε−u⟩+

∫Ω


)dt

→∫Q

g(u−u)− χ·∇(u−u) dxdt−∫ T

0

⟨∂u∂t

, u−u⟩dt for ε → 0, (8.217)

where χ is a weak limit (of a subsequence) of |∇uε|p−2∇uε in Lp′(Q;Rn). For the

convergence (8.217), we have used (8.214) and (8.215), which allows for

limε→0

∫Q

ε∇2uε :∇2u− ∂uε

∂t(u−u) dxdt

= limε→0

∫ T

0

⟨εΔ2uε +

∂uε

∂t, u⟩+⟨∂u∂t

, uε

⟩dt−

∫Ω

uε(T )u(T )− |u0|2 dx

=

∫ T

0

⟨∂u∂t

, u⟩+⟨∂u∂t

, u⟩dt− ∥∥u(T )∥∥2

L2(Ω)+∥∥u0

∥∥2L2(Ω)

=

∫ T

0

⟨∂u∂t

, u−u⟩dt.

Eventually, pushing u → u in Lp(I;W 1,p0 (Ω)) makes the final expression in (8.217)

arbitrarily closed to 0, which yields ‖∇uε‖Lp(Q;Rn) → ‖∇u‖Lp(Q;Rn) and then

uε → u in Lp(I;W 1,p0 (Ω)) by the uniform convexity of the space Lp(Q;Rn).


Exercise 8.87 (Conservation law regularized by Δ). Consider

∂u

∂t+ div(F (u))− εΔu = g, u|t=0 = u0, u|Σ = 0, (8.218)

where F : R → Rn has at most linear growth, i.e. |F (r)| ≤ C1 +C2|r|, and ε > 0.Make the basic estimates.79 Assuming also that F is Lipschitz continuous, makean estimate of u in W 1,2(I;L2(Ω))∩L∞(I;W 1,2

0 (Ω)).80 Prove further a bound foru in W 1,∞(I;L2(Ω)) ∩W 1,2(I;W 1,2

0 (Ω)).81 For estimation of the term div(F (u))on the left-hand side, see Exercise 9.27 below. For a special case n = 1 = ε andF (r) = 1

2r2, consider the so-called (regularized) Burgers equation

∂u

∂t+ u

∂u

∂x− ∂2u

∂x2= g in Q := (0, T )×(0, 1), u|x=0,1 = 0, u|t=0 = u0. (8.219)

Assuming 0 ≤ g ≤ K and u0 ∈ W 1,2(Ω), u0 ≥ 0, prove u ∈ L∞(I;W 1,2(Ω)) ∩W 1,2(I;L2(Ω))∩L2(I;W 2,2(Ω)) and 0 ≤ u(t, x) ≤ tK + ‖u0‖L∞(0,1).

82 Using thisregularity, prove also that the solution is unique.83

79Hint: Test by u gives

1

2

d

dt

∥∥u∥∥2L2(Ω)

+ ε∥∥∇u

∥∥2L2(Ω;Rn)

≤(C1

√measn(Ω) + C2

∥∥u∥∥L2(Ω)

)∥∥∇u∥∥L2(Ω;Rn)

,

so that by Young’s and Gronwall’s inequalities one obtains u bounded in L∞(I;L2(Ω)) ∩L2(I;W 1,2

0 (Ω)). Then the “dual” estimate of ∂∂t

u in L2(I;W−1,2(Ω)) follows standardly.80Hint: Test by ∂

∂tu gives∥∥∥∂u

∂t

∥∥∥2L2(Ω)

+1

ε2d

dt

∥∥∇u∥∥2L2(Ω;Rn)

≤ supr∈R

∣∣F ′(r)∣∣ ∥∥∇u

∥∥L2(Ω;Rn)

∥∥∥∂u∂t

∥∥∥L2(Ω)

.

Then use the Young and the Gronwall inequalities.81Hint: Differentiating (8.218) in time and testing by ∂

∂tu gives:

1

2

d

dt

∥∥∥∂u∂t

∥∥∥2

L2(Ω)+ ε

∥∥∥∇∂u

∂t

∥∥∥2

L2(Ω;Rn)≤ sup

r∈R

|F ′(r)|∥∥∥∂u∂t

∥∥∥L2(Ω)

∥∥∥ ∂

∂t∇u

∥∥∥L2(Ω;Rn)

.

Then use again the Young and the Gronwall inequalities.82Hint: First, test (8.219) by u to get u ∈ L2(I;W 1,2(Ω)) ∩ L∞(I;L2(Ω)); note that∫ 1

0u2 ∂

∂xu dx = 1

3

∫ 10

∂∂x

u3 dx = 0. Then test (8.219) by u− to get u ≥ 0. Then put w = u−Kt

to get ∂∂t

w + (w + Kt) ∂∂x

w − ∂2

∂x2 w = g − K ≤ 0 and test it by (w − ‖u0‖L∞(0,1))+ to get

w ≤ ‖u0‖L∞(0,1). Eventually, test (8.219) by ∂∂t

u and use u ∈ L∞(Q) to estimate

∥∥∥∂u∂t

∥∥∥2L2(0,1)

+1

2

d

dt

∥∥∥∂u

∂x

∥∥∥2L2(0,1)

= −∫ 1

0u∂u

∂x

∂u

∂tdx ≤ ∥∥u∥∥

L∞(Q)

∥∥∥∂u∂x

∥∥∥L2(0,1)

∥∥∥∂u∂t

∥∥∥L2(0,1)

to get u ∈ L∞(I;W 1,2(Ω)) ∩W 1,2(I;L2(Ω)). Then from (8.219) read u ∈ L2(I;W 2,2(Ω)).83Hint: Denoting u12 := u1 − u2, test the difference of (8.219) for u1 and u2 by u12:

1

2

d

dt

∥∥u12

∥∥2

L2(0,1)+

∥∥∥∂u12

∂x

∥∥∥2

L2(0,1)=

∫ 1

0

(u2

∂u2

∂x− u1

∂u1

∂x

)u12 dx

≤∥∥∥∂u1

∂x

∥∥∥L∞(0,1)

∥∥u12

∥∥2L2(0,1)

+1

2

∥∥u2

∥∥2L∞(0,1)

∥∥u12

∥∥2L2(0,1)

+1

2

∥∥∥∂u1

∂x

∥∥∥2L2(0,1)

.


Exercise 8.88 (Allen-Cahn equation [15]84). Consider the initial-boundary-valueproblem for the semilinear equation

∂u

∂t−Δu+ (u2 − 1)u = 0, u(0, ·) = u0,

∂u

∂ν

∣∣∣Σ= 0. (8.220)

Prove existence of a solution by Rothe’s or Galerkin’s method, deriving a-prioriestimates of the type ‖u‖L2(I;W 1,2(Ω))∩L4(Q) or ‖u‖L∞(I;W 1,2(Ω)∩L4(Ω)).

Exercise 8.89 (Cahn-Hilliard equation [89]85). Consider the initial-boundary-valueproblem for the semilinear 4th-order equation

∂u

∂t−Δβ(u) + Δ2u = g, u(0, ·) = u0, u|Σ =

∂u

∂ν

∣∣∣Σ= 0. (8.221)

Prove existence of a solution by Rothe’s or Galerkin’s method, deriving a-priori es-timates of u in W 1,2,2(I;W 2,2

0 (Ω),W−2,2(Ω))∩L∞(I;L2(Ω)), assuming β : R → Rsmooth of the form β = β1+β2 with β1 nondecreasing and β2 Lipschitz continuousand g ∈ L2(I;L2∗∗′

(Ω)).86

Exercise 8.90 (Viscous Cahn-Hilliard equation [189]87). Consider the semilinear4th-order pseudoparabolic equation

∂(u−Δu)

∂t−Δβ(u) + Δ2u = g, u(0, ·) = u0, u|Σ =

∂u

∂ν

∣∣∣Σ= 0 (8.222)

with β qualified as in Exercise 8.89, and modify the a-priori estimates therein.

Exercise 8.91 (Non-Newtonean fluids88). Analogously to (6.26a), consider

∂u

∂t− div σ

(e(u)

)+ (u · ∇)u +∇π = g, div u = 0, (8.223)

84Up to a suitable scaling, this equation is related to Ginzburg-Landau phase transition theory;for more details see e.g. Alikakos, Bates [14], Caginalp [87], Cahn, Hilliard [88], Hoffmann, Tang[205, Chap.2], Ohta, Mimura, and Kobayashi [317]. Let us remark that the full Ginzburg-Landausystem is related to superconductivity and received great attention in physics, being reflectedalso by Nobel prizes to L.D. Landau in 1962 and (1/3) to V.L. Ginzburg in 2003.

85This equation has been proposed to model isothermal phase separation in binary alloys ormixtures. There is an extensive spool of related references, e.g. Artstein and Slemrod [20] orElliott and Zheng [136], Novic-Cohen [315, 316], or von Wahl [422].

86Hint: Test (8.221) by u, use β′1 ≥ 0 and Gagliardo-Nirenberg’s inequality (Theorem 1.24

with q = p = r = k = 2, β = 1, λ = 1/2) to estimate

1

2

d

dt

∥∥u∥∥2L2(Ω)

+∥∥Δu‖2

L2(Ω)+

∫Ωβ′1(u)|∇u|2dx =

∫Ωgu− β′

2(u)|∇u|2dx

≤ ∥∥g∥∥L2∗∗′

(Ω)

∥∥u∥∥L2∗∗ (Ω)

+ sup β′2(·)

∥∥∇u∥∥2L2(Ω;Rn)

≤ ∥∥g∥∥L2∗∗′

(Ω)

∥∥u∥∥L2∗∗ (Ω)

+ CGN sup |β′2(·)|

∥∥u∥∥L2(Ω)

∥∥∇2u∥∥L2(Ω;Rn×n) ,

and then use still∫Ω|Δu|2dx =

∫Ω|∇2u|2dx under the considered boundary conditions, cf. Ex-

ample 2.46, and eventually Gronwall’s and Young’s inequalities.87This equation has been proposed by Grinfeld and Novick-Cohen [189] to model phase sepa-

ration in glass and polymer systems.88See Ladyzhenskaya [248] or Malek et al. [268, Sect.5.4.1]. Cf. also [81, 123] for p > 2n/(n+2).


with u|Σ = 0, u(0, ·) = u0 ∈ L2(Ω;Rn), e(u) as in (6.26a), and σ(e) = |e|p−2e;hence (6.28a,b) holds. Testing a Galerkin approximation of (8.223) by the approx-imate solution itself, prove existence of a weak solution if p is large enough.89

Exercise 8.92 (Semi-implicit time discretization). Consider p = 2 and the semilin-ear parabolic problem (8.167) with the linearization

[B(w, u)

](v) :=

∫Ω ∇u · ∇v+

|w|q1−2uv dx +∫Γ|w|q2−2uv dS and the semi-implicit formula (8.55), which leads

to⟨∂uτ

∂t, v⟩+

∫Ω

∇uτ ·∇v+ |uR

τ |q1−2uτv− gτv dx =

∫Γ

hτv− |uR

τ |q2−2uτv dS (8.224)

for a.a. t∈I and all v∈W 1,2(Ω) with uRτ defined in (8.202). Make the basic a-priori

estimate90 and prove the convergence for τ → 0.91

8.9 Global monotonicity approach, periodic problems

Sometimes, other methods can be applied on the abstract level, too. Let usmention a “global approach” which can solve the Cauchy problem directly onW 1,p,p′

(I;V, V ∗) provided V ⊂ H and which can straightforwardly be adapted for

89Hint: Using the identity∫Ω(u ·∇)u ·udx = 0, cf. (6.36), the suggested test gives bounds of u

in L∞(I;L2(Ω;Rn)) ∩ Lp(I;W 1,p0,div(Ω;Rn)), for W 1,p

0,div(Ω;Rn) see (6.29). For the dual estimate

of ∂∂t

u in Lp′(I;W 1,p0,div(Ω;Rn)∗), use Green’s Theorem 1.31 for the convective term:

∫Q(u · ∇)u · v dxdt = −

∫Q(u · ∇)v · u dxdt ≤ C

∥∥u∥∥2Lp� (Q;Rn)

∥∥∇v∥∥Lp(Q;Rn×n)

,

which needs 2/p� + 1/p ≤ 1. In view of (8.131), identify that p ≥ 11/5 (resp. p ≥ 2) is neededfor n = 3 (resp. n = 2). Make it more rigorous by using seminorms arisen in Galerkin’s method.

Alternatively, without using Green’s theorem, prove an estimate of ∂∂t

u in (Lp(I;W 1,p0,div(Ω;Rn))∩

L∞(I;L2(Ω)))∗ to be used as suggested in Remark 8.12.90Hint: Testing by uk

τ gives

1

2

∥∥ukτ

∥∥2

L2(Ω)− 1

2

∥∥uk−1τ

∥∥2L2(Ω)

+ τ∥∥∇uk

τ

∥∥2L2(Ω;Rn)

≤∫Ωgkτu

kτ − |uk−1

τ |q1−2|ukτ |2 dx

+

∫Γhkτu

kτ − |uk−1

τ |q2−2|ukτ |2 dS ≤

∥∥gkτ∥∥Lp∗′(Ω)

∥∥ukτ

∥∥Lp∗(Ω)

+∥∥hk

τ

∥∥Lp#

′(Γ)

∥∥ukτ

∥∥Lp# (Γ)

.

Then proceed as in (8.27) to get the bound in L∞(I;L2(Ω)) ∩ L2(I;W 1,2(Ω)). Further, thestrategy of Example 8.63(2) leads to the bound of ∂

∂tuτ in L2(I;W 1,2(Ω)∗) and also of ∂

∂tuτ in

M (I;W 1,2(Ω)∗).91Hint: By Corollary 7.9 with the interpolation (8.128), realize that, for a subsequence, uτ → u

in L2�−ε(Q), 2� = 2 + 4/n, ε > 0; cf. also (8.152). Since uRτ inherits all a-priori estimates as

uτ , also uRτ → u in L2�−ε(Q). Realize that these limits must indeed coincide with each other

because ‖uRτ − uτ‖L2(I;W1,2(Ω)∗) = O(τ) just by a modification of (8.40) with the boundedness

of { ddt

uτ}0<τ≤τ0 in L2(I;W 1,2(Ω)∗). The strategy for the traces uτ |Σ and uRτ |Σ is as in (8.153).

Then make the limit passage directly in (8.224) integrated over I.

8.9. Global monotonicity approach, periodic problems 289

periodic problems of the form

du

dt+A

(t, u(t)

)= f(t) for a.a. t∈I, u(0) = u(T ) . (8.225)

Obviously, considering A : R× V → V ∗ and f : R → V ∗ periodic with the periodT , a solution u to d

dtu + A (u) = f having the same given period can be justconstructed by a periodic prolongation of the solution u : I → V to (8.225); thisis why we refer to (8.225) as the periodic problem and u(0) = u(T ) as a periodiccondition.

Considering a mapping L : dom(L) → Lp′(I;V ∗), dom(L) ⊂ Lp(I;V ), the

following property of L will play an important role: for any w ∈ Lp′(I;V ∗) and

u ∈ Lp(I;V ):(∀v∈dom(L) :

⟨w−L(v), u−v

⟩ ≥ 0)

⇒{u∈dom(L),w = L(u).

(8.226)

A monotone mapping L satisfies (8.226) if92 and only if93 it is maximal monotone.The base of the direct method is the following observation:

Lemma 8.93 (Maximal monotonicity of ddt). Let L : u �→ d

dtu : dom(L) →Lp′

(I;V ∗) and either

dom(L) :={u∈W 1,p,p′

(I;V, V ∗); u(0) = u0

}(8.227)

for u0 ∈ H fixed, or

dom(L) :={u∈W 1,p,p′

(I;V, V ∗); u(0) = u(T )}. (8.228)

Then L is monotone, radially continuous, and satisfies (8.226).

Proof. 94 The monotonicity of L follows from the fact that, for any u, v ∈ dom(L),by using (7.22), we have

⟨L(u)−L(v), u−v

⟩=

∫ T

0

⟨d(u−v)

dt, u−v

⟩dt

=1

2

∥∥u(T )−v(T )∥∥2H− 1

2

∥∥u(0)−v(0)∥∥2H

{ ≥ 0 in case (8.227),= 0 in case (8.228),

because ‖u(0)− v(0)‖H = ‖u0 − u0‖H = 0 in case (8.227) and ‖u(0)− v(0)‖H =‖u(T )− v(T )‖H in case (8.228).

92Supposing the contrary (i.e. (8.226) does not hold for some (u, w)), we can derive thatGraph(L) ∪ {(u, w)} would be a graph of a monotone mapping larger than Graph(L), i.e. L isnot maximal monotone.

93Realize that, supposing w �= L(u), Graph(L) ∪ {(u, w)} would be a graph of a monotoneoperator, contradicting the fact that L is maximal.

94See, e.g., Barbu [37, p.167] (only the case (8.227)), Gajewski et al. [168, Sect. VI.1.2], Zeidler[427, Sect. 32.3b] for u0 = 0.


The radial continuity now means that 〈L(u+εv), v〉 = ∫ T

0 〈 ddt (u+εv), v〉dt →∫ T

0 〈 ddtu, v〉dt =〈L(u), v〉 for all v such that u + εv ∈ dom(L) for some (and thus

all) ε �= 0, which is obvious.For (8.227), let us first assume u0 ∈ V while, for (8.228), u0 can be considered

arbitrary from V , e.g. u0 = 0. Then let us take z ∈ V and ϕ ∈ C1(I) such thatϕ(0) = ϕ(T ) = 0, and put v(t) = ϕ(t)z + u0. Thus v ∈ dom(L). Obviously,ddtv = ϕ′z. Using the premise in (8.226), we have

0 ≤ ⟨w − L(v), u− v

⟩=⟨w, u

⟩+⟨L(v), v

⟩− ⟨w, v⟩ − ⟨L(v), u⟩=⟨w, u

⟩+

1

2‖v(T )‖2H − 1

2‖v(0)‖2H −

∫ T

0

⟨wϕ+ ϕ′u, z

⟩dt− ⟨w, u0

⟩=⟨w, u

⟩− ⟨∫ T

0

(wϕ + ϕ′u

)dt, z

⟩− ⟨w, u0

⟩, (8.229)

where v(T ) = u0 = v(0) has been used and where 〈w, u0〉 means∫ T

0〈w(t), u0〉dt.

Note also that the first integral in (8.229) is the Lebesgue one while the second

is a Bochner one. Since z ∈ V is arbitrary, it must hold that∫ T

0 ϕ′(t)u(t) +ϕ(t)w(t) dt = 0. As ϕ is arbitrary, it must hold that d

dtu = w in the sense ofdistributions.

Moreover, since w ∈ Lp′(I;V ∗), we have u ∈ W 1,p,p′

(I;V, V ∗). To proveu ∈ dom(L) we have to show u(0) = u0 or u(0) = u(T ), respectively.

Using the premise in (8.226) with w = ddtu and the by-part formula (7.22),

we have

0 ≤⟨dudt

− dv

dt, u− v

⟩=

1

2

∥∥u(T )− v(T )∥∥2H− 1

2

∥∥u(0)− v(0)∥∥2H. (8.230)

In case (8.227) if u0 ∈ V is considered, we can set v(t) = ((T−t)u0 + tuεT )/T with

uεT ∈ V chosen in such a way that uε

T → u(T ) in H ; note that u(T ) has a goodsense only in H due to Lemma 7.3 but not in V in general. From (8.230), we thenhave

0 ≤ ∥∥u(T )− uεT

∥∥2H− ∥∥u(0)− u0

∥∥2H

→ −∥∥u(0)− u0

∥∥2H

(8.231)

hence u(0) = u0. In case (8.228), from (8.230) we get

0 ≤ 1

2

∥∥u(T )∥∥2H− 1

2

∥∥u(0)∥∥2H− ⟨u(T ), v(T )⟩+⟨u(0), v(0)⟩+1

2

∥∥v(T )∥∥2H− 1

2

∥∥v(0)∥∥2H

=1

2

∥∥u(T )∥∥2H− 1

2

∥∥u(0)∥∥2H+⟨v(0), u(0)− u(T )

⟩(8.232)

because v(0) = v(T ). As v(0) is arbitrary, we get u(0) = u(T ).If u0 ∈H\V , we must replace the constant u0 which then does not belong

to Lp(I;V ) by some nonconstant u0 ∈ W 1,p,p′(I;V, V ∗) and then to choose v(t) =

ϕ(t)z + u0(t). Assuming u0(0) = u0, we have still v ∈ dom(L). Assuming still

u0(0) = u0(T ), it causes only the additional term∫ T

0 〈 ddt u0, u〉dt emerging in


(8.229). By choosing again z and then also ϕ arbitrary, one can argue exactly inthe same way as before.95 �

Remark 8.94 (Other conditions). Lemma 8.93 explains why the initial or theperiodic conditions are natural. E.g., if one would choose dom(L) := {u ∈W 1,p,p′

(I;V, V ∗); u(T ) = uT }, then L would not be monotone; i.e. prescribinga terminal condition u(T ) = uT does not yield a well-posed problem. In casedom(L) := {u∈W 1,p,p′

(I;V, V ∗); u(0) = u0, u(T ) = uT}, L would be monotonebut not maximal monotone; i.e. prescribing both the terminal and the initial con-ditions does not yield a well-posed problem, either. On the other hand, it is aneasy exercise to prove a modification of Lemma 8.93 for the so-called anti-periodiccondition u(0) + u(T ) = 0.

Let us now prove an abstract result, abbreviating V = Lp(I;V ). Let usalso assume that V can be approximated by finite-dimensional subspaces Vk suchthat, for u0 = 0, Vk ⊂ dom(L), Vk ⊂ Vk+1, and

⋃k∈N Vk is dense in dom(L) with

respect to the norm ‖u‖dom(L) = ‖u‖V +‖Lu‖V ∗ ; note that this space is separableso that such a chain of subspaces does exist.96 Note also that we, in fact, assumedL linear and that, as dom(L) �= V , we cannot use directly the Browder-MintyTheorem 2.18 for L+ A .

Lemma 8.95 (Surjectivity of L+A ). Let A : V → V ∗ be radially continuousand monotone, and L : dom(L) → V ∗ be affine and radially continuous97 andsatisfy (8.226). Moreover, let V admit the approximation by finite-dimensionalsubspaces in the above sense, and let L+A be coercive with respect to the norm ofV . Then L+A is surjective. Moreover, if A is strictly monotone, then (L+A )−1 :V ∗ → dom(L) does exist.

95Such an approximation can be made as follows: after a possible re-normalization of thereflexive space V so that both V and V ∗ are strictly convex (by Asplund’s theorem), we take u0

on I/2 = [0, T/2] as the solution to the initial-value problem ddt

u0 + Jp(u0) = 0 and u0(0)=u0

with Jp:V→V ∗ induced by the potential 1p‖ · ‖pV ; as for Jp cf. Example 8.103 below. By using

the test with u0, one has the estimate ‖u0‖Lp(I/2;V ) ≤ C. Then one can estimate

∥∥∥du0

dt

∥∥∥Lp′(I/2;V ∗)

= sup‖v‖Lp(I/2;V )≤1

∫ T/2

0

⟨Jp(u0), v

⟩dt

≤ sup‖v‖Lp(I/2;V )≤1

∫ T/2

0

∥∥u0

∥∥p−1

V

∥∥v∥∥Vdt ≤ ∥∥u0

∥∥p−1

Lp(I/2;V )≤ C.

On [T/2, T ], we define u0(t) := u0(T−t) to have u0(0) = u0 = u0(T ) and u0∈W 1,p,p′(I; V, V ∗).96As we consider V = Lp(I;V ) and L = d

dt, we get the norm on dom(L) identical with that in-

duced from W 1,p,p′(I;V, V ∗). An example of Vk can be span{ϕv; v∈Vk , ϕ∈C([0, T ]) a polynomof the degree ≤ k, ϕ(0) = 0} with Vk from (2.7) in case of the initial-value problem. For theperiodic problem, ϕ(0) = 0 is to be replaced by ϕ(0) = ϕ(T ).

97In fact, the assertion holds even without the requirement of the affinity and the radial-continuity assumptions about L.


Proof. 98 Take w0 ∈ dom(L).99 Then, for L(u) := L(u + w0)− L(w0), L is linearand dom(L) = dom(L)−w0 is a linear subspace. We put still A (u) := A (u+w0);note that again A is monotone and radially continuous and L + A is coercive.The equation L(u) + A (u) = f is equivalent with Lu+ A (u) = f − L(w0), theirsolutions being related to each other by u + w0 = u. Thus we can assume that Lis a linear operator without any loss of generality.

Consider Vk a finite-dimensional subspace of V as assumed, and endow Vk

by the norm of V . Denoting Ik : Vk → V the canonical inclusion, I∗k : V ∗ → V ∗k

and the norm of Ik and I∗k is at most 1. We will show that

∃uk∈Vk ∀v∈Vk :⟨Luk + A (uk), v

⟩= 〈f, v〉. (8.233)

Let us consider the mapping Bk : u �→ I∗k (Lu+A (u)) : Vk → V ∗k , which is radially

continuous, monotone, and also coercive due to the estimate

〈Bk(u), u〉‖u‖V

=〈I∗k (Lu+ A (u)), u〉

‖u‖V=

〈Lu+ A (u), Iku〉‖u‖V

=〈Lu+ A (u), u〉

‖u‖V≥ a

(‖u‖V

)→ +∞ (8.234)

for ‖u‖V → ∞, u ∈ Vk. By the Browder-Minty Theorem 2.18, the equationBk(u) = I∗kf has a solution uk. Such uk satisfies also (8.233) and, from (8.234),

a(‖uk‖V

) ≤ 〈Bk(uk), uk〉‖uk‖V

≤ ∥∥Bk(uk)∥∥

V ∗k

=∥∥I∗kf∥∥V ∗

k

≤ ∥∥I∗k∥∥L (V ∗,V ∗k )‖f‖V ∗ =

∥∥Ik∥∥L (Vk,V )‖f‖V ∗ ≤ ‖f‖V ∗ (8.235)

hence {uk}k∈N is bounded in V , and then also, by the monotonicity of L and by(8.233) and (8.235),⟨

A (uk), uk

⟩ ≤ ⟨Luk, uk

⟩+⟨A (uk), uk

⟩=⟨I∗kf, uk

⟩=⟨f, Ikuk

⟩=⟨f, uk

⟩ ≤ ∥∥f∥∥V ∗∥∥uk

∥∥V

≤ ∥∥f∥∥V ∗a

−1(‖f‖V ∗

). (8.236)

To conclude that {A (uk)}k∈N is bounded in V ∗, as in (2.42), for any ε > 0, weestimate∥∥A (uk)

∥∥V ∗ ≤ 1

εsup

‖v‖V ≤ε

(⟨A (uk), uk

⟩+⟨A (v), v

⟩ − ⟨A (v), uk

⟩). (8.237)

Now we use that {〈A (uk), uk〉}k∈N is bounded due to (8.236), {〈A (v), v〉; ‖v‖V ≤ε} is bounded if ε > 0 is small because A is locally bounded around the origin

98See Gajewski [168, Section III.2.2]. Alternatively, one can approximate the operator L+ Aby adding the duality mapping J .

99If u0 ∈ V , we can take simply w0(t) = u0. If u0 ∈ H \V , we can take w0 ∈ W 1,p,p′(I; V, V ∗),e.g., a solution of the initial-value problem d

dtw0 + Jp(w0) � 0 with w0(0) = u0 like in the proof

of Lemma 8.93.


due to Lemma 2.15, and eventually {〈A (v), uk〉}k∈N is bounded by (8.235) if ε issmall.

In view of the above a-priori estimates, we can consider some (u, χ) ∈ V ×V ∗

being the limit of a subsequence such that uk ⇀ u and A (uk) ⇀ χ. Furthermore,take z ∈ V , v ∈ dom(L) and vk ∈ Vk such that vk → v with respect to the norm‖ · ‖dom(L). Then, by (8.233), the monotonicity of L (written between vk and uk),and the monotonicity of A (written between z and uk), we have

0 ≤ ⟨Lvk−Luk, vk−uk

⟩+⟨A (z)−A (uk), z−uk

⟩=⟨Lvk−Luk−A (uk), vk−uk

⟩+⟨A (z), z−uk

⟩− ⟨A (uk), z−vk⟩

=⟨Lvk − f, vk − uk

⟩+⟨A (z), z − uk

⟩− ⟨A (uk), z − vk⟩. (8.238)

Now we can pass to the limit with k → ∞. Note that L is continuous with respectto the norm ‖ · ‖dom(L) so that 〈Lvk, vk〉 → 〈Lv, v〉. Thus we come to

0 ≤ ⟨Lv − f, v − u⟩+⟨A (z), z − u

⟩− ⟨χ, z − v⟩

=⟨Lv − f + χ, v − u

⟩+⟨A (z)− χ, z − u

⟩, (8.239)

which now holds for any v ∈ dom(L) and any z ∈ V . Choosing z := u in (8.239),we obtain ⟨

Lv − f + χ, v − u⟩ ≥ 0 (8.240)

for any v ∈ dom(L). This implies u ∈ dom(L) and Lu = f − χ because L satisfies(8.226). Knowing u ∈ dom(L), we can also choose v := u in (8.239), which gives⟨

A (z)− χ, z − u⟩ ≥ 0 (8.241)

for any z ∈ V . Since A is monotone and radially continuous, by the Minty trick(see Lemma 2.13) we obtain A (u) = χ. Altogether, Lu+A (u) = (f −χ)+χ = f .

If A is strictly monotone, so is L+A and thus the solution to the equationLu+ A (u) = f is unique, which means that (L+ A )−1 is single-valued. �

Theorem 8.96 (Existence). Let the Caratheodory mapping A : I×V → V ∗ satisfythe growth condition (8.80) with C constant and A(t, ·) be radially continuous,monotone and semi-coercive in the sense of (8.95) with Z := V but with | · |V :=‖ · ‖V and, in case (8.228), with c2 = 0. Then both the Cauchy problem (8.1)and the periodic problem (8.225) have solutions. Moreover, if A(t, ·) is strictlymonotone for a.a. t ∈ I, these solutions are unique.

Proof. First, as in the proof of Lemma 8.95, we can consider L linear withoutloss of generality. Since A is a Caratheodory mapping satisfying the growth con-dition (8.80) with C constant, A maps Lp(I;V ) into Lp′

(I;V ∗) and A is radiallycontinuous, cf. Example 8.52.

Directly from (8.95) with c2 = 0 and | · |V := ‖ · ‖V , we get the coercivity ofL+A with respect to the norm of V on dom(L) from (8.228) simply by integration


over I:∫ T

0

⟨dudt

+A(t, u), u⟩dt ≥

∫ T

0

1

2

d

dt‖u‖2H + c0‖u(t)‖pV − c1(t)‖u(t)‖V dt

≥ 1

2‖u(T )‖2H − 1

2‖u(0)‖2H+

∫ T

0

(c0−ε)‖u(t)‖pV − Cεcp′1 (t) dt

≥ (c0 − ε)‖u‖pLp(I;V )− Cε‖c1‖p′

Lp′(I). (8.242)

In case of (8.227), c2 > 0 can be admitted by modifying (8.242) by estimating∫ t1

0

⟨dudt

+A(t, u), u⟩dt ≥

∫ t1

0

1

2

d

dt‖u‖2H + c0‖u‖pV − c1‖u‖V − c2‖u‖2H dt

≥ 1

2‖u(t1)‖2H − 1

2‖u0‖2H+

∫ t1

0

(c0−ε)‖u‖pV − Cεcp′1 − c2‖u‖2H dt

for any t1 ∈ I, from which one obtains

(c0−ε)‖u‖pLp(I;V ) ≤(1

2‖u0‖2H +

∫ T

0

⟨dudt

+A(t, u), u⟩+ Cεc

p′1

)e∫ T0

c2 dt

by using the Gronwall inequality (1.66), so that∫ T

0

⟨dudt

+A(t, u), u⟩dt ≥ (c0−ε)e−‖c2‖L1(I)

∥∥u∥∥pLp(I;V )

− 1

2

∥∥u0

∥∥2H− Cε

∥∥c1∥∥p′

Lp′(I),

which shows the coercivity L+ A on dom(L) from (8.227).Then we use Lemmas 8.93 and 8.95 for V = Lp(I;V ) and L defined in

Lemma 8.93; note that then dom(L) = W 1,p,p′(I;V, V ∗). �

Remark 8.97. If p < 2, the coercivity of L+A on dom(L) from (8.228) obviouslyfails for (8.95) with c2 > 0. Also, the uniqueness for (8.225) fails if A(t, ·) is merelymonotone, while for (8.1) the monotonicity of A(t, ·) is sufficient for the uniqueness,as we saw in Theorem 8.34. This is because L + A is then strictly monotone ondom(L) from (8.227) but not on dom(L) from (8.228).

Exercise 8.98 (Limit passage for ∂u∂t −Δpu). Modify Example 8.66 by exploiting

monotonicity of ∂u∂t −Δpu instead of the monotonicity of −Δpu only.

8.10 Problems with a convex potential: direct method

For φ : V → R convex, let us define the conjugate function φ∗ : V ∗ → R by

φ∗(v∗) := supv∈V

〈v∗, v〉 − φ(v). (8.243)

The transformation φ �→ φ∗ is called the Legendre transformation in the smoothcase, or the Legendre-Fenchel transformation in the general case.

8.10. Problems with a convex potential: direct method 295

φ∗(v∗)

φ

1

v

v∗

Figure 18. An illustration of a convex φ and thevalue of its conjugate φ∗ at a given v∗.

Always, φ∗ is convex, lower semicontinuous, and φ∗(v∗) + φ(v) ≥ 〈v∗, v〉,which is called Fenchel’s inequality [148]. Also φ∗∗ = φ if and only if φ is lowersemicontinuous; recall that V is here considered as reflexive. If φ is lower semicon-tinuous and proper (i.e. φ > −∞ and φ �≡ +∞), then100

v∗ ∈ ∂φ(v) ⇔ v ∈ ∂φ∗(v∗) ⇔ φ∗(v∗) + φ(v) = 〈v∗, v〉. (8.244)

For f ∈ V ∗, we have [φ− f ]∗(v∗) = φ∗(v∗ + f) because

[φ− f ]∗(v∗) = supv∈V

〈v∗, v〉 − (φ(v)− 〈f, v〉)= sup

v∈V〈v∗ + f, v〉 − φ(v) = φ∗(v∗ + f). (8.245)

For any constant c, one has [φ+ c]∗ = φ∗ − c. Also, [cφ]∗ = cφ∗(·/c) provided c ispositive because[

cφ]∗(v∗) = sup

v∈V〈v∗, v〉 − cφ(v) = c

(supv∈V

⟨v∗c, v⟩− φ(v)

)= cφ∗

(v∗c

). (8.246)

Furthermore, φ∗1 ≤ φ∗2 if φ1 ≥ φ2. Moreover, φ smooth implies φ∗ strictly con-vex.101

Suppose now that A(t, u) = ϕ′u(t, u) for some potential ϕ : I × V → R suchthat ϕ(t, ·) : V → R is Gateaux differentiable for a.a. t ∈ I with ϕ′u(t, ·) : V → V ∗

denoting its differential. Again, we consider the situation V ⊂ H ⊂ V ∗ for aHilbert space H . Let us define Φ : W 1,p,p′

(I;V, V ∗) → R by

Φ(u) :=1

2‖u(T )‖2H +

∫ T

0

ϕ(t, u(t)) + ϕ∗(t, f(t)− du

dt

)− 〈f(t), u(t)〉dt, (8.247)

where ϕ∗(t, ·) is conjugate to ϕ(t, ·). Let us mention that Φ is well-defined providedϕ is a convex Caratheodory integrand102 satisfying

c‖u‖pV ≤ ϕ(t, u) ≤ C(1 + ‖u‖pV

), (8.248)

100The inclusion v∗ ∈ ∂φ(v) is equivalent to 〈v∗ , u〉 − φ(u) ≤ 〈v∗, v〉 − φ(v) holding for anyu ∈ V , which is equivalent to φ∗(v∗) = supu∈U 〈v∗, u〉−φ(u) = 〈v∗, v〉−φ(v), from which alreadythe equivalence of the first and the third statements follows. If φ is lower semicontinuous, thenφ∗∗ = φ, so that the equivalence with the second statement follows by symmetry.101Suppose a contrary, i.e. Graph(φ∗) contains a segment, then ∂φ(u) is not a singleton for u,which contradicts Gateaux differentiability of ϕ, cf. Exercise 5.34.102This means that ϕ(t, ·) : V → R is convex and continuous while ϕ(·, v) : I → R is measurable.


which implies the analogous estimates for the conjugate ϕ∗, namely

(cp)1−p′

p′‖u∗‖p′

V ∗ ≥ ϕ∗(t, u∗) ≥ (Cp)1−p′

p′‖u∗‖p′

V ∗ − C; (8.249)

cf. Example 8.103. Note that (8.248) ensures that Nϕ : Lp(I;V ) → L1(I) while

(8.249) ensures that ϕ∗(t, ·) has at most p′-growth so that Nϕ∗ : Lp′(I;V ∗) →

L1(I); the needed fact that ϕ∗ is a Caratheodory integrand can be proved fromseparability of V and from (8.249).103

Theorem 8.99 (Brezis-Ekeland variational principle [68]). Let ϕ be aCaratheodory function satisfying (8.248) and ϕ(t, ·) be convex and continuouslydifferentiable.104 Then:

(i) If u ∈ W 1,p,p′(I;V, V ∗) solves the Cauchy problem (8.1), then u minimizes Φ

over dom(L) from (8.227) and, moreover, Φ(u) = 12‖u0‖2H .

(ii) Conversely, if Φ(u) = 12‖u0‖2H for some u ∈ dom(L) from (8.227), then u

minimizes Φ over dom(L) and solves the Cauchy problem (8.1).

Proof. By (8.245) we have ϕ∗(t, f + ·) = [ϕ(t, ·)− f ]∗, and therefore, by using theFenchel inequality and (7.22), we have always

Φ(u) =1

2‖u(T )‖2H +

∫ T

0

ϕ(t, u(t)) − ⟨f(t), u(t)⟩+ ϕ∗(t, f(t)− du

dt

)dt

≥∫ T

0

−⟨dudt

, u(t)⟩dt+

1

2‖u(T )‖2H =

1

2‖u0‖2H (8.250)

for any u ∈ dom(L). If u solves the Cauchy problem (8.1), i.e. ddtu+ ϕ′u(t, u(t)) =

f(t) or, in other words, − ddtu = (ϕ(t, u) − 〈f(t), u〉)′u, then, by using (8.245) and

(8.244),

[ϕ− f(t)

](u(t)

)+[ϕ− f(t)

]∗(− du

dt

)= ϕ

(t, u(t)

)− ⟨f(t), u(t)⟩+ ϕ∗(t, f(t)− du

dt

)= −

⟨dudt

, u(t)⟩. (8.251)

Hence this u attains the minimum of Φ on dom(L), proving thus (i).

103For a countable dense set {vk}k∈N ⊂ V , we have ϕ∗(t, v∗) = supk∈N〈v∗, vk〉 − ϕ(t, vk) andthen ϕ∗(·, v∗), being a supremum of a countable collection of measurable functions {〈v∗ , vk〉 −ϕ(·, vk)}k∈N, is itself measurable. Moreover, (8.249) makes the convex functional ϕ∗(t, ·) locallybounded from above on the Banach space V ∗, hence it must be continuous.104This guarantees, in particular, that A is a Caratheodory mapping: the continuity of A(t, ·) =ϕ′(t, ·) is just assumed while the measurability of A(·, u) = ϕ′(·, u) follows from the measurabilityof both ϕ(·, u+ εv) and ϕ(·, u) for any u, v ∈ V , hence by Lebesgue’s Theorem 1.14 〈A(·, u), v〉 =Dϕ(·, u; v) = limε→0

1εϕ(·, u+ εv)− 1

εϕ(·, u) is Lebesgue measurable, too, and eventually A(·, u)

itself is Bochner measurable by Pettis’ Theorem 1.34 by exploiting again the (generally assumed)separability of V .


Conversely, suppose that Φ(u) = 12‖u0‖2H . Note that, in view of (8.250),

u ∈ dom(L) then also minimizes Φ on dom(L). Moreover, by (8.250),

0 = Φ(u)− 1

2‖u0‖2H =

∫ T

0

ϕ(t, u(t)) + ϕ∗(t, f(t)−du

dt

)−⟨f(t)−du

dt, u(t)

⟩dt ≥ 0;

the last inequality goes from the Fenchel inequality. Thus, for a.a. t, it holds thatϕ(t, u(t)) − 〈f(t), u(t)〉 + ϕ∗(t, f(t) − d

dtu) + 〈 ddtu, u(t)〉 = 0. By (8.244), this is

equivalent with ddtu = f(t) − ϕ′u(t, u(t)), so that u ∈ dom(L) solves the Cauchy

problem (8.1). �

Corollary 8.100. Let the assumptions of Theorem 8.99 be fulfilled. Then the solu-tion to (8.1) is unique.

Proof. As ϕ is smooth, ϕ∗ is strictly convex, and thus also Φ is strictly convexbecause L is injective on dom(L) from (8.227). Thus Φ can have only one minimizeron the affine manifold dom(L). By Theorem 8.99(i), it gives uniqueness of thesolution to (8.1). �

Remark 8.101 (Periodic problems). Modification for periodic problem (8.225) uses:

Φ(u) :=∫ T

0 ϕ(t, u(t)) + ϕ∗(t, f(t) − ddtu) − 〈f(t), u(t)〉dt and dom(L) from (8.228).

The minimum of Φ on dom(L) is 0. Modification of Corollary 8.100 for periodicproblems requires ϕ(t, ·) strictly convex because L is not injective on dom(L) from(8.228) so that strict convexity of ϕ∗(t, ·) does not ensure strict convexity of Φ.

Note that Theorem 8.99(i) stated the existence of a minimizer of Φ ondom(L) by means of an a-priori knowledge that the solution to the Cauchyproblem (8.1) does exist. We can however proceed in the opposite way, whichgives us another (so-called direct) method to prove existence of a solution to(8.1). Note that Theorem 8.99(ii) does not imply this existence result becauseminu∈dom(L) Φ(u) = 1

2‖u0‖2H is not obvious unless we know that the solution to(8.1) exists.

Theorem 8.102 (Direct method). Let the assumptions of Theorem 8.99 be ful-filled and let also ϕ∗(t, ·) be smooth. Then:

(i) Φ attains its minimum on dom(L).

(ii) Moreover, this (unique) minimizer represents the solution to the Cauchy prob-lem (8.1).

Proof. (i) Φ is convex, continuous,W 1,p,p′(I;V, V ∗) is reflexive, and by Lemma 7.3

the mapping u �→ u(0) : W 1,p,p′(I;V, V ∗) → H is continuous so that dom(L) is

closed in W 1,p,p′(I;V, V ∗). Moreover, by (8.248) and (8.249), Φ is coercive on

dom(L): indeed, due to the lower bound

Φ(u) ≥∫ T

0

[c‖u‖pV − 〈f, u〉+ (Cp)1−p′

p′

∥∥∥f − du

dt

∥∥∥p′

V ∗− C

]dt, (8.252)


obviously Φ(u) → +∞ for ‖u‖Lp(I;V ) + ‖ ddtu‖Lp′(I;V ∗) → ∞. Then the existence

of a minimizer follows by the direct method.(ii) We must calculate Φ′ and then use simply 〈Φ′(u), v〉 = 0 for any v

belonging to the tangent cone to dom(L) at u, i.e for any v ∈ W 1,p,p′(I;V, V ∗)

with v(0) = 0.Without loss of generality, we can consider u0 = 0; cf. the proof of

Lemma 8.95. Then dom(L) is a linear subspace and, if endowed by the topol-ogy of W 1,p,p′

(I;V, V ∗), L is continuous and injective, and therefore L−1 doesexist on Range(L).

Denote ϕT (u) =∫ T

0ϕ(t, u(t)) dt and similarly ϕ∗T (ξ) =

∫ T

0ϕ∗(t, ξ(t)) dt. Note

that ϕ∗T : Lp′(I;V ∗) → R is conjugate to ϕT : Lp(I;V ) → R.105 Using L : u �→

ddtu : W 1,p,p′

(I;V, V ∗) → Lp′(I;V ∗) and (8.247), we can write

Φ(u) = [ϕT − f ](u) + ϕ∗T (f − L(u)) +1

2‖u(T )‖2H . (8.253)

Using the first equivalence in (8.244) and realizing that ∂ϕT = {ϕ′T } and ∂ϕ∗T ={[ϕ∗T ]′}, we obtain [ϕ∗T ]

′ = [ϕ′T ]−1. In particular, denoting w := [ϕ∗T ]

′(f − L(u)),we have w = [ϕ′T ]

−1(f − L(u)), so that

ϕ′T (w) = f − L(u). (8.254)

We can then calculate106

Φ′(u) = ϕ′T (u)− f − L∗([ϕ∗T ]

′(f − L(u)))+ u(T ) · δT

= ϕ′T (u)− f + L(w) + (u(T )− w(T )) · δT (8.255)

where δT : W 1,p,p′(I;V, V ∗) → H : u �→ u(T ) and where we used also the identity

L∗(w) = −L(w) + w(T ) · δT which follows from the by-part formula for w ∈W 1,p,p′

(I;V, V ∗) ⊂ Lp′(I;V ∗)∗ if v(0) = 0 is taken into account:

⟨L(w), v

⟩=⟨dwdt

, v⟩= −

⟨w,

dv

dt

⟩+⟨w(T ), v(T )

⟩− ⟨w(0), v(0)⟩=⟨w,L(v)

⟩+⟨w(T ), v(T )

⟩=⟨L∗(w), v

⟩+⟨w(T ), v(T )

⟩. (8.256)

From 〈Φ′(u), v〉 = 0 with v vanishing on [0, T − ε] and such that v(T ) = u(T )−w(T ), passing also ε → 0, we obtain from (8.255) that 0 = 〈v(T ), u(T )−w(T )〉 =‖u(T )− w(T )‖2H , i.e.

u(T ) = w(T ). (8.257)

105This follows from the identity [ϕT ]∗(ξ) = supu∈Lp(I;V )

∫ T0 〈ξ(t), u(t)〉 − ϕ(t, u(t)) dt =∫ T

0 supu∈V [〈ξ(t), u〉 − ϕ(t, u)]dt = ϕ∗T (ξ) which can be proved by a measurable-selection tech-

nique.106We also use the formula ∂Φ(A(u)) = A∗∂Φ(u) which holds provided 0 ∈ int(Range(A) −Dom(Φ)).


Furthermore, taking v with a compact support in (0, T ), we get from 〈Φ′(u), v〉 = 0with Φ′ in (8.255) that

ϕ′T (u)− f + L(w) = 0. (8.258)

Subtracting (8.254) and (8.258), testing it by u − w, and using monotonicity ofϕ′T , we get

0 =⟨L(u)− L(w), u − w

⟩+⟨ϕ′T (w) − ϕ′T (u), u− w

⟩ ≤ 1

2

d

dt

∥∥u− w∥∥2H. (8.259)

Using (8.257) and the Gronwall inequality backward, we get u = w. Putting thisinto (8.258) (or alternatively into (8.254)), we get d

dtu + ϕ′u(t, u(t)) = f(t); herewe use also that107 [ϕ′T (u)](t) = ϕ′u(t, u(t)). As u ∈ dom(L), the initial conditionu(0) = u0 is satisfied, too. �

Example 8.103. For φ(v) = 1p‖v‖pV , the conjugate function is φ∗(v∗) = 1

p′ ‖v∗‖p′

V ∗ ,

which follows by the Holder inequality and which explains why p′ := p/(p−1) has

been called a conjugate exponent. This also implies (c‖ · ‖pV )∗ = cp 1p′ ‖ ·cp‖p

′V ∗ =

(cp)1−p′

p′ ‖ · ‖p′V ∗ .

Example 8.104 (Parabolic evolution by p-Laplacean108). Considering V =W 1,p

0 (Ω), ϕ(t, u) =∫Ω

1p |∇u|pdx and 〈f(t), u〉 =

∫Ωg(t, x)u(x) dx corresponds,

in the variant of the Cauchy problem, to the initial-boundary-value problem

∂u

∂t− div

(|∇u|p−2∇u)

= g in Q,

u = 0 on Σ,

u(0, ·) = u0 in Ω;

⎫⎪⎪⎬⎪⎪⎭ (8.260)

cf. Example 4.23. Let us abbreviate Δp : W 1,p0 (Ω) → W−1,p′

(Ω) the p-Laplacean,this means Δpu := div(|∇u|p−2∇u). One can notice that ϕ(u) = 1

p‖u‖pW 1,p0 (Ω)

provided ‖u‖W 1,p0 (Ω) := ‖∇u‖Lp(Ω;Rn). Then ϕ∗(ξ) = 1

p′ ‖ξ‖p′

W−1,p′(Ω). Moreover,

one can see109 that Δpu = −Jp(u) where Jp : V → V ∗ is the duality mapping withrespect to the p-power defined by the formulae 〈Jp(u), u〉 = ‖Jp(u)‖V ∗‖u‖V and

‖Jp(u)‖V ∗ = ‖u‖p−1V . Hence, ‖ξ‖V ∗ = ‖J−1

p (ξ)‖p−1V implies here ‖ξ‖p′

W−1,p′(Ω)=

‖Δ−1p ξ‖p

W 1,p0 (Ω)

so that

ϕ∗(ξ) =1

p′∥∥ξ∥∥p′

W−1,p′ (Ω)=

1

p′∥∥Δ−1

p ξ∥∥pW 1,p

0 (Ω)=

1

p′∥∥∇Δ−1

p ξ∥∥pLp(Ω;Rn)

. (8.261)

107See e.g. [209, Theorem II.9.24].108For the linear case (i.e. p = 2) see Brezis and Ekeland [68] or also Aubin [26].109Cf. Proposition 3.14 which, however, must be modified. Note that, for p = 2, Jp = J with Jthe standard duality mapping (3.1).


It yields the following explicit form of the functional Φ:

Φ(u) =

∫ T

0

(∫Ω

1

p|∇u|p + 1

p′

∣∣∣∇Δ−1p

(g − ∂u

∂t

)∣∣∣p− gu dx+⟨∂u∂t

, u⟩)

dt. (8.262)

We can observe that the integrand in (8.262) is nonlocal in space; some nonlocality(in space or in time) is actually inevitable as shown by Adler [5] who provedthat there is no local variational principle yielding (8.260) as its Euler-Lagrangeequation.


Further reading concerning evolution by pseudomonotone mappings can includemonographs by Brezis [65], Gajewski et al. [168], Lions [261], Ruzicka [376,Sect. 3.3.5-6], Showalter [383, Chap.III], Zeidler [427, Vol.II B]. Beside Rothe’soriginal artical [356], also e.g. [304] and special monographs by Kacur [219] andRektorys [347] are devoted to Rothe’s method. Moreover, various semi-implicitmodifications of the basic Rothe method leading to efficient numerical schemeshas been devised in [214, 221, 222].

Galerkin’s method is in a special monograph Thomee [405], and also in Zeidler[427, Sect.30].

Quasilinear parabolic equations are thoroughly exposed in Ladyzhenskaya,Solonikov, Uraltseva [249], Liebermann [259, Chap.13], Lions [261, Chap.II.1], andTaylor [402, Chap.15]. Semilinear equations received special attentions in Henri[200], Pao [324] and Robinson [351]. Monotone parabolic equations are also inWloka [424]. The weak solution we derived in Theorems 8.13 and 8.31 on anabstract level for weakly continuous mappings can in concrete cases be derived forquasilinear equations, too; cf. [268, Sect.5.3]. Fully nonlinear equations of the type∂∂tu+ a(Δu) = g (also not mentioned in here) are, e.g., in Dong [126, Chap.9,10],or Liebermann [259, Chap.14–15].

Regularity theory for parabolic equations is exposed, e.g., in Bensoussan andFrehse [50], Kacur [219, Chap.3], Ladyzhenskaya et al. [249], Lions and Magenes[262], and Taylor [402, Chap.15]. For further advanced topics in parabolic problemssee DiBenedetto [120], Galaktionov [169], and Zheng [429]. In the context of theiroptimal control, we refer e.g. to Fattorini [144, Part II] or Troltzsch [410].

The Navier-Stokes equations have been thoroughly exposed by Constantinand Foias [106], Feistauer [147], Lions [261, Chap.I.6], Sohr [391], Taylor [402,Chap.17], and Temam [403]. Generalization for non-Newtonean fluids is in La-dyzhenskaya [248] and Malek et al. [268].

For time periodic problems we refer to Vejvoda et al. [415]. The methodused for the existence Theorem 8.96 applies also to the general case when Ais pseudomonotone, see Brezis [65] or also Zeidler [427, Section 32.4]. See alsoLions [261, Section 7.2.2] for A being the mapping of type (M). The anti-periodicproblems have been addresed e.g. in [8, 198].


The Brezis-Ekeland principle in the weaker version as in Theorem 8.99, in-vented in [68] and independently by Nayroles [301] and thus sometimes addressedas a Brezis-Ekeland-Nayroles principle, can be found even for nonsmooth prob-lems in Aubin and Cellina [28, Section 3.4] or Aubin [26] for V a Hilbert space,f = 0, and autonomous systems. The improvement as a direct method, i.e. The-orem 8.102, is from [362]. Newer investigations are by Ghoussoub [185] and Tzou[186], Stefanelli [396], and Visintin [419, 420], cf. also (11.54) below. Other varia-tional principles for parabolic equations had been surveyed by Hlavacek [203].

Chapter 9

Evolution governed by accretivemappings

Now we replace the weak compactness and monotonicity method by the normtopology technique and a completeness argument. Although, in comparison withthe former technique, this method is not the basic one, it widens in a worthwhileway the range of the monotone-mapping approach presented in Chapter 8.

Again we consider the Cauchy problem (8.4) but now with A : dom(A) → Xan m-accretive mapping (or, more generally, A+λI m-accretive for some λ ≥ 0),X a Banach space whose norm will be denoted by ‖ · ‖ as in Chap. 3, dom(A)dense1 in X , f ∈L1(I;X), u0∈X .

9.1 Strong solutions

Let us agree to call u ∈ W 1,1(I;X) ≡ W 1,∞,1(I;X,X) a strong solution to theinitial-value problem (8.4) if {A(u(t))}t≥0 is bounded in X , hence in particularu(t) ∈ dom(A) for all t∈I, and (8.4) is valid a.e. on I := [0, T ], as well as the initialcondition in (8.4) holds, and the distributional derivative d

dtu ∈ L1(I;X) is alsothe weak derivative, i.e. w-limε→0

1εu(t + ε) − 1

εu(t) for a.a. t∈ I.2 The followingassertion will be found useful:

Lemma 9.1 (Chain rule). Let Φ : X → R be convex and locally Lipschitz con-tinuous, and u ∈W 1,1(I;X) have also the weak derivative. Then Φ ◦ u : I → Ris a.e. differentiable and d

dtΦ(u(t)) = 〈f, ddtu(t)〉 with any f ∈ ∂Φ(u(t)) holds for

a.a. t∈I.

1In fact, if cl(dom(A)) �= X, we must require u0∈cl(dom(A)) in Lemma 9.4 and Theorem 9.5.2In general, u∈W 1,1(I;X) need not have the weak derivative, but if it has, then it coincides

with the distributional derivative ddt

u.


303

304 Chapter 9. Evolution governed by accretive mappings

Proof. As u ∈ W 1,1(I;X) is bounded and absolutely continuous and Φ locallyLipschitz, Φ ◦ u is absolutely continuous, and hence a.e. differentiable. Considert∈I at which Φ◦u as well as u have (weak) derivatives. As Φ is convex, Φ(u(t+ε)) ≥Φ(u(t)) + 〈f, u(t+ε)−u(t)〉 for any f ∈∂Φ(u(t)) and any ε ∈ [−t, T − t], cf. (5.2)for v := u(t+ε). In particular, for ε > 0, we obtain

Φ(u(t+ε))− Φ(u(t))

ε≥⟨f,

u(t+ε)− u(t)

ε

⟩and (9.1a)

Φ(u(t))− Φ(u(t−ε))

ε≤⟨f,

u(t)− u(t−ε)

ε

⟩. (9.1b)

Passing to the limit in (9.1a,b), we obtain respectively ddtΦ(u(t)) ≥ 〈f, d

dtu(t)〉 andddtΦ(u(t)) ≤ 〈f, d

dtu(t)〉. �

The following assertion justifies the definition of the strong solution:

Proposition 9.2 (Uniqueness, continuous dependence). Let Aλ, defined by

Aλ := A+ λI (9.2)

with I : X → X denoting the identity, be accretive, and X∗ be separable. Then thestrong solution, if it exists, is unique. Moreover,

‖u1 − u2‖C(I;X) ≤ emax(λ,0)T(‖f1 − f2‖L1(I;X) + ‖u01 − u02‖

)(9.3)

with ui being the unique strong solutions toddtui+A(ui) = fi, ui(0) = u0i, i = 1, 2.

Proof. Our strategy is to test the difference ddt (u1−u2)+A(u1)−A(u2) = f1−f2 by

J(u1−u2). Using Lemma 9.1 for Φ = 12‖ · ‖2 and that ∂Φ = J , cf. Example 5.2, we

obtain 〈j∗, ddt (u1−u2)〉 = 1

2ddt‖u1−u2‖2 for any j∗∈J(u1−u2) a.e. on I. As we have

the liberty in taking j∗ ∈J(u1−u2) arbitrarily, we select it so that 〈j∗, Aλ(u1) −Aλ(u2)〉 ≥ 0 a.e. on I, using the accretivity of Aλ. Then, for a.a. t∈I, we have

‖u1 − u2‖ d

dt‖u1 − u2‖ =

1

2

d

dt‖u1 − u2‖2

≤ 1

2

d

dt‖u1−u2‖2 +

⟨j∗, Aλ(u1)−Aλ(u2)

⟩=⟨j∗,

d(u1−u2)

dt+Aλ(u1)−Aλ(u2)

⟩=⟨j∗, f1−f2 + λ(u1−u2)

⟩≤ ‖j∗‖∗

(‖f1−f2‖+ λ‖u1−u2‖)= ‖u1−u2‖

(‖f1−f2‖+ λ‖u1−u2‖), (9.4)

then we divide3 it by ‖u1−u2‖, which gives ddt‖u1−u2‖ ≤ ‖f1−f2‖+λ‖u1−u2‖, and

use the Gronwall inequality (1.66). This gives ‖u1(t)−u2(t)‖ ≤ eλt(‖u01−u02‖ +∫ t

0 ‖f1(ϑ)−f2(ϑ)‖e−λϑdϑ). The uniqueness comes as a side product. �3This step is legal. Indeed, assume that, at some t > 0, it holds that ‖u1−u2‖ = 0 and

simultaneously ddt‖u1−u2‖ > ‖f1−f2‖ + λ‖u1−u2‖ ≥ 0, which would however imply existence

of ε > 0 such that ‖u1−u2‖(t− ε) = ‖u1−u2‖(t) + o(ε) < 0, a contradition.

9.1. Strong solutions 305

The existence of a solution will be proved by the Rothe method, based againon the recursive formula (8.5) combined with (8.58). The Rothe functions uτ anduτ are again defined by (8.6) and (8.7), respectively.

Lemma 9.3 (Existence of Rothe’s sequence). Let Aλ be m-accretive, f ∈L1(I;X), u0 ∈ X. Then uτ does exist provided τ < 1/λ (or τ arbitrary if λ ≤ 0).

Proof. We have [I+ τA

](u1

τ ) = u0 +

∫ τ

0

f(t) dt ∈ X, (9.5)

which has at least one solution u1τ ∈ dom(A) because I + τA = I+ τAλ − τλI =

(1−τλ)[I+ τ1−τλAλ] and Aλ is m-accretive and thus [I+λ1Aλ] is surjective for any

λ1 := τ/(1− τλ) positive, in particular for any τ > 0 sufficiently small, obviouslyτ < 1/λ (if λ > 0), cf. Definition 3.4 and (9.2).

Recursively, we obtain u2τ , u

3τ , etc. �

Lemma 9.4 (A-priori estimates). Let Aλ be m-accretive, f ∈ L1(I;X), andu0 ∈ X. Then, for τ < 1/λ, it holds that

‖uτ‖C(I;X) ≤ C, ‖uτ‖L∞(I;X) ≤ C. (9.6)

If, in addition, f ∈ W 1,1(I;X) and u0 ∈ dom(A), then also∥∥∥duτ

dt

∥∥∥L∞(I;X)

≤ C(‖f‖W 1,1(I;X) + ‖A(u0)‖

). (9.7)

Proof. First, by making a transformation as in the proof of Lemma 8.95, one canassume dom(A) � 0.

The identity (8.5) with (8.58) can be rewritten into the form

ukτ =

[1τI+Aλ

]−1(1τuk−1τ + fk

τ + λukτ

), where fk

τ :=1

τ

∫ kτ

(k−1)τ

f(t) dt. (9.8)

As Aλ is accretive, [I + τAλ]−1 is non-expansive (see Lemma 3.7) and therefore

[ 1τ I + Aλ]−1 is Lipschitz continuous with the constant τ . Denoting vτ := [ 1τ I +

Aλ]−1(0), we obtain from (9.8) that ‖uk

τ − vτ‖ ≤ τ‖( 1τ uk−1τ + fk

τ +λukτ )− 0‖, from

which we obtain

‖ukτ‖ ≤ ‖uk−1

τ ‖+ τ(‖fk

τ ‖+ λ‖ukτ‖)+ ‖vτ‖. (9.9)

Then we get the estimates (9.6) by using the discrete Gronwall inequality (1.70)4

because τλ < 1 and because ‖vτ‖ = O(τ); indeed,

‖vτ‖ = ‖vτ−0‖ ≤∥∥∥(vτ+τAλ(vτ )

)− (0+τAλ(0))∥∥∥ = τ‖Aλ(0)‖ = O(τ) (9.10)

4Note that the condition τ < 1/a in (1.70) reads here just as τ < 1/λ.


because [I+ τAλ]−1 is non-expansive and because vτ+τAλ(vτ ) = 0.

Furthermore, assuming f ∈ W 1,1(I;X) and u0 ∈ dom(A), we apply J(ukτ −

uk−1τ ) to (8.5). We get, by using accretivity of Aλ,

5

1

τ‖uk

τ−uk−1τ ‖2 ≤

⟨J(uk

τ−uk−1τ ),

ukτ−uk−1

τ

τ

⟩+⟨J(uk

τ−uk−1τ ), Aλ(u

kτ )−Aλ(u

k−1τ )

⟩=⟨J(uk

τ − uk−1τ ), fk

τ −A(uk−1τ ) + λ(uk

τ−uk−1τ )

⟩=⟨J(uk

τ − uk−1τ ), fk−1

τ −A(uk−1τ ) + (fk

τ − fk−1τ ) + λ(uk

τ−uk−1τ )

⟩=⟨J(uk

τ − uk−1τ ),

uk−1τ − uk−2

τ

τ+ (fk

τ − fk−1τ ) + λ(uk

τ−uk−1τ )

⟩≤ ∥∥J(uk

τ−uk−1τ )

∥∥∗

(∥∥∥uk−1τ −uk−2

τ

τ

∥∥∥+ ∥∥fkτ −fk−1

τ

∥∥+ λ∥∥uk

τ−uk−1τ

∥∥). (9.11)

Dividing it by ‖ukτ − uk−1

τ ‖ = ‖J(ukτ − uk−1

τ )‖∗, for k ≥ 2 we get∥∥∥ukτ−uk−1

τ

τ

∥∥∥ ≤∥∥∥uk−1

τ −uk−2τ

τ

∥∥∥+ τ∥∥∥fk

τ −fk−1τ

τ

∥∥∥+ λτ∥∥∥uk

τ−uk−1τ

τ

∥∥∥, (9.12)

which can be treated by Gronwall’s inequality (1.70) if λτ < 1, by using alsosummability of the second right-hand-side term in (9.12) uniformly in τ if f ∈W 1,1(I;X).6 For k = 1, similarly as in (9.11), we obtain ‖u1

τ−u0τ‖2 ≤ τ〈J(u1−u0),

f1τ−A(u0

τ )+λ(u1τ−u0

τ )〉 ≤ τ‖u1τ−u0

τ‖ ‖f1τ−A(u0

τ )+λ(u1τ−u0

τ )‖, from which further∥∥∥u1τ−u0

τ

τ

∥∥∥ ≤ ∥∥f1τ −A(u0

τ ) + λ(u1τ−u0

τ )∥∥

≤ ∥∥f(0)∥∥+ ∥∥A(u0)∥∥+ τ

(∥∥f∥∥W 1,1(I;X)

+ λ∥∥∥u1

τ−u0τ

τ

∥∥∥). (9.13)

From this, exploiting also u0 ∈ dom(A), (9.7) follows. �Theorem 9.5 (Existence of strong solutions, Kato [225, 226]). Let X∗

be uniformly convex,7 Aλ be m-accretive for some λ ≥ 0, f ∈ W 1,1(I;X), andu0∈dom(A). Then there is u∈W 1,∞(I;X) such that uτ → u in C(I;X) and thisu is a strong solution to (8.4).

Proof. We show that {uτ}τ>0 is a Cauchy sequence in C(I;X). For τ, σ > 0,we have d

dtuτ + A(uτ ) = fτ and ddtuσ + A(uσ) = fσ. Subtracting them, testing

5If J is set-valued, we must again select a suitable element from J to ensure non-negativity of the second left-hand-side term, while the required identity ‖uk

τ−uk−1τ ‖2 =

〈J(ukτ−uk−1

τ ), ukτ−uk−1

τ 〉 holds always.6Assuming f ∈ W 1,1(I;X), one must modify (8.75)–(8.76) to bound

∑T/τk=1 ‖fk

τ − fk−1τ ‖

independently of τ .7A counterexample by Webb (cf. [118, Sect.14.3, Example 6]) shows that this assumption is

indeed essential.

9.1. Strong solutions 307

it by J(uτ − uσ), and using also accretivity of Aλ and monotonicity8 of J (seeLemma 3.2(ii)), we obtain

1

2

d

dt

∥∥uτ−uσ

∥∥2 ≤⟨J(uτ−uσ),

d

dt(uτ−uσ)

⟩+⟨J(uτ−uσ), Aλ(uτ )−Aλ(uσ)

⟩=⟨J(uτ−uσ), fτ−fσ

⟩+⟨J(uτ−uσ)− J(uτ−uσ),

d

dt(uτ−uσ)

⟩+ λ

⟨J(uτ−uσ), uτ−uσ

⟩ ≤ ∥∥J(uτ−uσ)∥∥∗∥∥fτ−fσ

∥∥+ λ∥∥uτ−uσ

∥∥2≤ 1

2

∥∥uτ − uσ

∥∥2 + 1

2

∥∥fτ − fσ∥∥2 + λ

∥∥uτ − uσ

∥∥2,and the term ‖uτ − uσ‖ can further be estimated by∥∥uτ−uσ

∥∥ ≤ ∥∥uτ−uσ

∥∥+ ∥∥uτ−uτ

∥∥+ ∥∥uσ−uσ

∥∥ =∥∥uτ−uσ

∥∥+O(τ) +O(σ) (9.14)

due to the estimate∥∥uτ − uτ

∥∥L∞(I;X)

≤ τ∥∥∥ d

dtuτ

∥∥∥L∞(I;X)

≤ τC (9.15)

where (9.7) was used. We eventually obtain

d

dt

∥∥uτ−uσ

∥∥2 ≤ ∥∥uτ−uσ

∥∥2 + ∥∥fτ−fσ∥∥2 + 2λ

∥∥uτ−uσ

∥∥2 + O(max(τ, σ)2). (9.16)

Using ‖fτ − fσ‖ → 0 in L2(I) for τ, σ → 0, by the Gronwall inequality, we get‖uτ − uσ‖C(I;X) → 0 if τ, σ → 0; more precisely, ‖uτ − uσ‖C(I;X) = O(max(τ, σ)).Since C(I;X) is complete, the limit of the sequence {uτ}τ>0 exists.

Take t ∈ I. Thanks to (9.7) and f ∈ W 1,1(I;X) ⊂ L∞(I;X), A(uτ (t)) isbounded so that we can assume that A(uτ (t)) converges weakly in X to somew(t); here we used reflexivity of X∗ (and hence also of X) by Milman-Pettis’theorem. Simultaneously, uτ (t) → u(t) because uτ (t) → u(t) and uτ (t)−uτ (t) → 0,cf. (9.15). Now we have to show that the graph of the m-accretive mapping Aλ

is (norm×weak)-closed. Exploiting the assumed uniform convexity of X∗, hencecontinuity of J , cf. Lemma 3.2(iii), we have J(v−uτ (t)) → J(v−u(t)), and thus⟨

J(v−u(t)), Aλ(v)− wλ

⟩= lim

τ→0

⟨J(v−uτ (t)), Aλ(v)−Aλ(uτ (t))

⟩ ≥ 0 (9.17)

for any v ∈ dom(Aλ) and for wλ := w(t) + λu(t). As I + Aλ is surjective, wemay choose v ∈ V so that v + Aλ(v) = u(t) + wλ, and then from (9.17) weget 0 ≤ 〈J(v − u(t)), Aλ(v) − wλ〉 = 〈J(v − u(t)), u(t) − v〉 = −‖u(t) − v‖2.Hence u(t) = v ∈ dom(Aλ) and, from v + Aλ(v) = u(t) + wλ, we further obtainu(t) +Aλ(u(t)) = u(t) + wλ, i.e. wλ = Aλ(u(t)) and thus also w(t) = A(u(t)).

Moreover, fτ → f in L1(I;X) and also ddtuτ ⇀ d

dtu weakly in L2(I;X).

Altogether, we can pass to the limit in the equation ddtuτ +A(uτ ) = fτ , obtaining

8By monotonicity of J , it holds that〈J(uτ − uσ)− J(uτ − uσ),ddt

(uτ − uσ)〉 ≤ 0.


(8.4). Since uτ (0) = u0 and uτ → u in C(I;X), the initial condition u(0) = u0 issatisfied, too. By Milman-Pettis’ theorem, the uniformly convex X∗ is reflexive,and hence so is X . Therefore the distributional derivative d

dtu is also the weak

derivative; note that u, having the distributional derivative ddtu in L1(I;X), is

also absolutely continuous due to the estimate ‖u(t) − u(s)‖ ≤ ∫ t

s ‖ ddϑu‖ dϑ,9 so

that, by Komura’s Theorem 1.39, ddtu is even the strong derivative. Eventually,

{A(u(t))}t∈I = {f(t)− ddtu}t∈I is bounded in X , as required. �

9.2 Integral solutions

We define u ∈ C(I;X) the integral solution (of type λ) if u(0) = u0 and

∀v∈dom(A), 0 ≤ s ≤ t ≤ T :1

2‖u(t)− v‖2 ≤ 1

2‖u(s)− v‖2

+

∫ t

s

⟨f(ϑ)−A(v), u(ϑ) − v

⟩s+ λ

∥∥u(ϑ)− v∥∥2dϑ ; (9.18)

where λ refers to the accretivity of Aλ := A+λI and where 〈u, v〉s := sup⟨u, J(v)

⟩is the semi-inner product, cf. (3.7). Note that integral solutions need not rangeover dom(A) and their time derivative need not exist, in contrast with the strongsolutions.

Proposition 9.6 (Consistency of definition (9.18)). Let Aλ := A+λI be ac-cretive (for a sufficiently large λ). Any strong solution is the integral solution.

Proof. Using accretivity of Aλ and the properties (3.1) of the duality mapping J ,we get the following calculations:10

1

2‖u(t)− v‖2 − 1

2‖u(s)− v‖2 =

∫ t

s

d

dϑ

1

2‖u(ϑ)− v‖2dϑ

=

∫ t

s

⟨J(u(ϑ)−v),

du

dϑ

⟩dϑ =

∫ t

s

⟨J(u(ϑ)−v),

du

dϑ− f(ϑ) +A(v)

⟩+⟨J(u(ϑ)− v), f(ϑ)−A(v)

⟩dϑ

≤∫ t

s

⟨J(u(ϑ)−v),−A(u(ϑ)) +A(v)

⟩+⟨f(ϑ)−A(v), u(ϑ)− v

⟩sdϑ

≤∫ t

s

⟨J(u(ϑ)−v),−Aλ(u(ϑ)) +Aλ(v)

⟩9This can be seen from (7.2) used for ϕ(ϑ) :=

∫ ts�ε(ϑ − θ)dθ with �ε from (7.11), and by

passing to the limit with ε → 0, which gives u(t) − u(s) =∫ ts

ddϑ

u dϑ at all Lebesgue points sand t of u, cf. Theorem 1.35,

10If J is set-valued, we must choose a suitable j∗ ∈ J(u(ϑ)− v) in order to guarantee the non-positiveness of the first right-hand side term and use that the identity d

dt12‖u(ϑ)−v‖2 = 〈j∗, du

dt〉

holds a.e. for whatever choice j∗ ∈ J(u(ϑ)− v) we made, cf. Lemma 9.1.

9.2. Integral solutions 309

+λ⟨J(u(ϑ)−v), u(ϑ)− v

⟩+⟨f(ϑ)−A(v), u(ϑ) − v

⟩sdϑ

≤∫ t

s

λ‖u(ϑ)− v‖2 + ⟨f(ϑ)−A(v), u(ϑ)− v⟩sdϑ . (9.19)

�

Hence, Theorem 9.5 yields an integral solution if the data f and u0 are regularenough and X is reflexive with X∗ uniformly convex. We put the regularity off and u0 off, and later in Theorem 9.9 we get rid also of the reflexivity of X .Even more important and here more difficult, is to prove uniqueness that showsselectivity of the definition of integral-solutions, which is not self-evident at all.

Theorem 9.7 (Existence and uniqueness). Let X∗ be uniformly convex, Aλ bem-accretive, f ∈ L1(I;X), and u0 ∈ cl dom(A), in particular just u0 ∈ X if A isdensely defined. Then (8.4) has a unique integral solution.

Proof. Take fε ∈ W 1,1(I;X) and u0ε ∈ dom(A) such that fε → f in L1(I;X) andu0ε → u0 in X . Denote uε ∈ C(I;X) the strong solution to the problem

duε

dt+A

(uε(t)

)= fε(t) , uε(0) = u0ε , (9.20)

obtained in Theorem 9.5, so that by (9.19) for any v ∈ dom(A) and any 0 ≤ s ≤t ≤ T it holds that

1

2‖uε(t)− v‖2 ≤ 1

2‖uε(s)− v‖2

+

∫ t

s

⟨fε(ϑ)−A(v), uε(ϑ)− v

⟩s+ λ‖uε(ϑ)− v‖2dϑ. (9.21)

By (9.3), {uε}ε>0 is a Cauchy sequence in C(I;X) which is complete, so that thereis some u ∈ C(I;X) such that uε → u in C(I;X).

Since uε(0) = u0ε → u0 and simultaneously uε(0) → u(0), we can see thatu(0) = u0. Moreover, passing to the limit in (9.21) and using the continuity of〈·, ·〉s,11 we can see that u is an integral solution to (8.4).

For uniqueness of the integral solution, let us consider, besides the just ob-tained integral solution u, some other integral solution, say u. Take uε the strongsolution corresponding to fε and u0ε as above. As uε(σ) ∈ dom(A) for arbitrary

11For the limit passage in the integral, we use Lebesgue’s Theorem 1.14 and realize that, byLemma 3.2(iii), J is continuous and thus so is (u, v) �→ 〈u, J(v)〉 = 〈u, v〉s, and that {fε}ε>0 canhave an integrable majorant, and 〈·, ·〉s has a linear growth in the left-hand argument, while forthe right-hand argument we have L∞-a-priori estimates (9.6) valid for f ’s in L1(I;X) and u0’sin X.


σ ∈ I, we thus can put a test element v := uε(σ) into (9.21), obtaining

1

2‖u(t)− uε(σ)‖2 − 1

2‖u(s)− uε(σ)‖2

≤∫ t

s

⟨f(ϑ)−A(uε(σ)), u(ϑ)− uε(σ)

⟩s+ λ‖u(ϑ)− uε(σ)‖2dϑ

≤∫ t

s

⟨f(ϑ)− fε(σ), u(ϑ)− uε(σ)

⟩sdϑ

+ λ

∫ t

s

‖u(ϑ)− uε(σ)‖2 dϑ+

∫ t

s

⟨duε

dt(σ), u(ϑ) − uε(σ)

⟩sdϑ, (9.22)

where we used A(uε(σ)) = fε(σ)− ddtuε(σ). Let us apply

∫ b

a dσ to (9.22). By usingFubini’s theorem, we can re-write the last integral as∫ b

a

[∫ t

s

⟨duε

dt(σ), u(ϑ)− uε(σ)

⟩sdϑ

]dσ

=

∫ t

s

[∫ b

a

⟨duε

dt(σ), u(ϑ)− uε(σ)

⟩sdσ

]dϑ

=1

2

∫ t

s

‖u(ϑ)− uε(a)‖2 − ‖u(ϑ)− uε(b)‖2 dϑ. (9.23)

Abbreviating ϕε(t, σ) :=12‖u(t) − uε(σ)‖2 and ψε(ϑ, σ) := 〈f(ϑ) − fε(σ), u(ϑ) −

uε(σ)〉s, we get∫ b

a

ϕε(t, σ)− ϕε(s, σ) dσ ≤∫ b

a

∫ t

s

ψε(σ, ϑ) + 2λϕε(ϑ, σ) dϑdσ

+

∫ t

s

ϕε(ϑ, a)− ϕε(ϑ, b) dϑ. (9.24)

Pass to the limit with ε → 0, using uε → u in C(I;X) (here the first part of thistheorem is exploited) and fε → f in L1(I;X) (with an integrable majorant), sothat in particular

lim supε→0

∫ b

a

ψε(ϑ, σ) dσ = lim supε→0

∫ b

a

⟨f(ϑ)− fε(σ), u(ϑ) − uε(σ)

⟩sdσ

≤∫ b

a

⟨f(ϑ)− f(σ), u(ϑ)− u(σ)

⟩sdσ =:

∫ b

a

ψ(ϑ, σ) dσ; (9.25)

again we used the upper semicontinuity of 〈·, ·〉s. Denoting naturally ϕ(ϑ, σ) :=12‖u(ϑ)− u(σ)‖2, we have eventually (9.24) without the subscript ε.

Now we need still to smooth ϕ and ψ for a moment. We can do it by convo-lution with a kernel like (7.11), cf. Figure 16(middle). For simplicity, here we use


1δχ[−δ/2,δ/2] with δ > 0, prolong u and u for t < 0 by continuity and f for t < 0

by zero, and define ϕδ(ϑ, σ) := 1δ2

∫ δ/2

−δ/2

∫ δ/2

−δ/2ϕ(ϑ − ξ, σ − ζ)dζdξ and similarly

ψδ(ϑ, σ) :=1δ2

∫ δ/2

−δ/2

∫ δ/2

−δ/2 ψ(ϑ− ξ, σ− ζ)dζdξ. Using (9.24) (without ε, of course)

we obtain∫ b

a

ϕδ(t, σ) − ϕδ(s, σ) dσ +

∫ t

s

ϕδ(ϑ, b)− ϕδ(ϑ, a) dϑ

=1

δ2

∫ δ/2

−δ/2

∫ δ/2

−δ/2

(∫ b

a

ϕ(t−ξ, σ−ζ)− ϕ(s−ξ, σ−ζ) dσ

+

∫ t

s

ϕ(ϑ−ξ, b−ζ)− ϕ(ϑ−ξ, a−ζ) dϑ

)dξdζ

=1

δ2

∫ δ/2

−δ/2

∫ δ/2

−δ/2

(∫ b−ζ

a−ζ

ϕ(t−ξ, σ)− ϕ(s−ξ, σ) dσ

+

∫ t−ξ

s−ξ

ϕ(ϑ, b−ζ)− ϕ(ϑ, a−ζ) dϑ

)dξdζ

≤ 1

δ2

∫ δ/2

−δ/2

∫ δ/2

−δ/2

(∫ b−ζ

a−ζ

∫ t−ξ

s−ξ

ψ(ϑ, σ) + 2λϕ(ϑ, σ)dϑdσ

)dξdζ

=1

δ2

∫ δ/2

−δ/2

∫ δ/2

−δ/2

(∫ b

a

∫ t

s

ψ(ϑ−ξ, σ−ζ)+2λϕ(ϑ−ξ, σ−ζ)dϑdσ

)dξdζ

=

∫ b

a

∫ t

s

ψδ(ϑ, σ) + 2λϕδ(ϑ, σ) dϑdσ (9.26)

for any δ/2 ≤ a ≤ b ≤ T − δ/2 and δ/2 ≤ s ≤ t ≤ T − δ/2. Thus we get (9.24)with δ instead of ε. As now ϕδ and ψδ are absolutely continuous, we can deduce

∂

∂ϑϕδ(ϑ, σ) +

∂

∂σϕδ(ϑ, σ) ≤ ψδ(ϑ, σ) + 2λϕδ(ϑ, σ) ; (9.27)

to see it, just apply∫ t

sdϑ∫ b

adσ to (9.27) to get (9.26). Putting ϑ = σ and denoting

ϕδ(ϑ) := ϕδ(ϑ, ϑ) and ψδ(ϑ) := ψδ(ϑ, ϑ), we obtain

d

dϑϕδ(ϑ) ≤ ψδ(ϑ) + 2λϕδ(ϑ). (9.28)

From Gronwall’s inequality, we get

ϕδ(t) ≤(ϕδ(0) +

∫ t

0

∣∣ψδ(ϑ)∣∣dϑ)e2λ+t (9.29)

for any t ∈ I. Now we can pass δ → 0. Obviously, ϕδ(t) =1δ2

∫ δ/2

−δ/2

∫ δ/2

−δ/2ϕ(t−ξ, t−

ζ)dζdξ → ϕ(t, t) = 12‖u(t)−u(t)‖2 for each t ∈ I. In particular, ϕδ(0) → 1

2‖u(0)−


u(0)‖2 = 12‖u0 − u0‖2 = 0. Moreover, using |ψ(ϑ, σ)| = |〈f(ϑ) − f(σ), u(ϑ) −

u(σ)〉s| ≤ C‖f(ϑ)− f(σ)‖ with C = ‖u‖C(I;X) + ‖u‖C(I;X), we obtain∫ t

0

∣∣ψδ(ϑ)∣∣ dϑ =

∫ t

0

1

δ2

∣∣∣ ∫ δ/2

−δ/2

∫ δ/2

−δ/2

ψ(ϑ− ξ, ϑ− ζ) dζdξ∣∣∣ dϑ

≤ C

δ2

∫ t

0

∫ δ/2

−δ/2

∫ δ/2

−δ/2

∥∥f(ϑ−ξ)− f(ϑ−ζ)∥∥ dζdξdϑ

≤∫ t

0

C

δ2

∫ δ/2

−δ/2

∫ δ/2

−δ/2

∥∥f(ϑ−ξ)− f(ϑ)∥∥+ ∥∥f(ϑ)− f(ϑ−ζ)

∥∥ dζdξdϑ≤ C

∫ t

0

1

δ

∫ δ/2

−δ/2

∥∥f(ϑ−ξ)− f(ϑ)∥∥ dξdϑ

+C

∫ t

0

1

δ

∫ δ/2

−δ/2

∥∥f(ϑ−ζ)− f(ϑ)∥∥ dζdϑ. (9.30)

The last two terms then converge to zero, cf. Theorems 1.14 and 1.35. Hence from(9.29) we get in the limit that ϕ(t, t) = 0, so u(t) = u(t) for any t∈I. �

Having the uniqueness of the integral solution, the stability (9.3) followsjust by the limit passage by strong solutions corresponding to regularized data(fiε, u0iε) → (fi, u0i), i = 1, 2. This yields:

Corollary 9.8 (Stability). Let the conditions of Theorem 9.7 hold. Then theestimate (9.3) holds for ui being the unique integral solution corresponding to thedata (fi, u0i) ∈ L1(I;X)× cl dom(A), i = 1, 2.

The requirement of the uniform convexity of X∗ (hence, in particular, reflex-ivity of X) we used in Theorem 9.7 can be restrictive in some applications butit can be weakened. We do it in the next assertion, proving thus existence of theintegral solution by the Rothe method combined with a regularization of data.

Theorem 9.9 (Existence: the nonreflexive case). Let f ∈ L1(I;X), u0 ∈cl dom(A), and Aλ be m-accretive for some λ. Then (8.4) has an integral solution.

Proof. We take the regularization fε ∈ W 1,1(I;X) and u0ε ∈ dom(A) as in theproof of Theorem 9.7, i.e. fε → f in L1(I;X) and u0ε → u0 in X , but here weadditionally assume∥∥fε∥∥W 1,1(I;X)

= O(1ε

)and

∥∥A(u0ε)∥∥ = O

(1ε

). (9.31)

Denote uετ ∈ C(I;X) the Rothe solution corresponding to fε and u0ε with a timestep τ > 0, i.e. it holds that d

dtuετ + A(uετ ) = (fε)τ , and uετ (0) = u0ε; here weused m-accretivity of Aλ. Then, combining (9.7) and (9.15),

‖uετ − uετ‖ = τO(1ε

)= O(

τ

ε). (9.32)


Let us define J1 : X → X∗ by 〈J1(v), v〉 = ‖J1(v)‖∗‖v‖ and ‖J1(v)‖∗ = 1 for v �= 0while ‖J1(v)‖∗ = 0 for v = 0; cf. Example 8.104 for Jp. Then we test the difference

of ddtuτε +A(uτε) = (fε)τ and d

dtuσδ +A(uσδ) = (fδ)σ by J1(uτε−uσδ). Realizing

that J1 has a potential ‖ · ‖ and applying Lemma 9.1, we get ddt‖uτε−uσδ‖ ≤

‖(fε)τ − (fδ)σ‖+λ‖uτε−uσδ‖ and by (9.32), like in (9.16), we get ddt‖uτε−uσδ‖ ≤

‖(fε)τ − (fδ)σ‖+ λ‖uτε−uσδ‖ + O(max( τε ,σδ )). By Gronwall’s inequality, we can

see that both {uτε}τ,ε>0, τ=o(ε) and {uτε}τ,ε>0, τ=o(ε) are Cauchy in C(I;X) withthe same limit in this complete space, say u.

As J has a potential 12‖ · ‖2, cf. Example 5.2, we have, as in (9.19), the

estimate

1

2‖uk

ετ − v‖2 − 1

2‖uk−1

ετ − v‖2 ≤ ⟨j∗, ukετ − uk−1

ετ

⟩=⟨j∗, uk

ετ − uk−1ετ − τfk

ετ + τA(v) + τ(fkετ −A(v))

⟩=⟨j∗,−τA(uk

ετ ) + τA(v) + τ(fkετ − A(v))

⟩= τ

⟨j∗, Aλ(v)−Aλ(u

kετ ) + fk

ετ −A(v) + λ(ukετ − v)

⟩≤ τ

⟨fkετ −A(v), uk

ετ − v⟩s+ τλ‖uk

ετ − v‖2, (9.33)

where the former inequality holds for any j∗ ∈ J(ukετ − v) while for the last one

we must select j∗ ∈ J(ukετ − v) suitably so that 〈j∗, Aλ(u

kετ )−Aλ(v)〉 ≥ 0.

Summing it between two arbitrary time levels, we get

‖uετ (t)−v‖22

≤ ‖uετ (s)−v‖22

+

∫ t

s

⟨(fε)τ (ϑ)−A(v), uετ (ϑ)−v

⟩s+ λ

∥∥uετ (ϑ)−v∥∥2dϑ

with t, s ∈ {kτ ; k = 0, . . . , T/τ}. Fixing t and s, let us now make the limit passagewith τ = T 2−k, k → ∞, ε → 0, τ = o(ε). The above inequality turns then into(9.18) using again the upper semicontinuity of 〈·, ·〉s and, thanks to τ = o(ε), alsousing Lemma 8.7.12 Altogether, we can see that u satisfies (9.18) for t and s froma dense subset of I. Then, by continuity, (9.18) holds for any t, s ∈ I.

Moreover, since uετ (0) = u0ε → u0 and also uετ (0) → u(0), we can see thatu(0) = u0. Hence u is an integral solution to (8.4). �

Remark 9.10 (Periodic problems13). If, in addition, Aλ is accretive for some λ < 0,then u0 �→ u(T ) is a contraction on X , namely

‖u1(T )− u2(T )‖ ≤ eλT ‖u01 − u02‖, (9.34)

12For the limit passage in the integral, we note that {(fε)τ}ε=o(τ)>0 can have an integrablemajorant, 〈·, ·〉s has a linear growth in the left-hand argument, while for the right-hand argumentuετ (·)−v we have L∞-a-priori estimates (9.6) valid for f ’s in L1(I;X) and u0’s in X. Then oneis to use the upper semicontinuity of 〈·, ·〉s, cf. Exercise 3.34 on p.112, and Fatou’s Theorem 1.15.

13See Barbu [37, Sect.III.2.2], Brezis [66, Sect.III.6], Crandall and Pazy [110], Showalter [383,Sect.IV,Prop.7.3], or Straskraba and Vejvoda [399], or Vainberg [414, Sect.VIII.26.4].


cf. Corollary 9.8.14 Having proved this contraction, one can use the Banach fixedpoint Theorem 1.12 to prove existence of a unique periodic integral solution,i.e. u ∈ C(I;X) satisfying (9.18) and the periodic condition u(T ) = u(0). Trivially,also the mapping u0 �→ −u(T ) is a contraction on X and thus one can prove exis-tence of a unique anti-periodic integral solution, i.e. u ∈ C(I;X) satisfying (9.18)and the anti-periodic condition u(T ) = −u(0).

Example 9.11 (Connection with the monotone-mapping approach). Consider A1 :V → V ∗ monotone, radially continuous, and bounded, A2 : H → H Lipschitzcontinuous with a Lipschitz constant , and A1+A2 : V → V ∗ coercive, V ⊂ H ∼=H∗ ⊂ V ∗, V being embedded into H densely and compactly. We define X andA : dom(A) → X by

X := H, dom(A) :={v∈V ; A1(v)∈H

}, A :=

(A1+A2

)∣∣dom(A)

. (9.35)

Then Aλ is accretive for λ ≥ because J : X = H ∼= H∗ = X∗ is the identity and⟨J(u−v), A(u)−A(v) + λ(u−v)

)⟩H∗×H

=(u−v,A1(u)−A1(v)

)+ λ‖u−v‖2H

+(u−v,A2(u)−A2(v)

) ≥ ⟨A1(u)−A1(v), u−v⟩V ∗×V

+ (λ− )‖u−v‖2H ≥ 0

(9.36)

for any u, v ∈ dom(A) and for λ ≥ . Moreover, as A2 : V → V ∗ is totallycontinuous, it is pseudomonotone as well as A1, so that I+A1 +A2 is also pseu-domonotone. As A1 + A2 is coercive, I + A1 + A2 is coercive, too. Thus, for anyf ∈ H , the equation u + A1(u) + A2(u) = f has a solution u ∈ V . Moreover,A1(u) = f − u − A2(u) ∈ H so that u ∈ dom(A). Hence I + A : dom(A) → H issurjective, so that Aλ is m-accretive. This approach gives additional informationabout solutions to d

dtu+A(u) = f , u(0) = u0, in comparison with Theorems 8.16

and 8.18. E.g. if f ∈ W 1,1(I;H) and u0 ∈ V such that A(u0) ∈ H , then ddtu exists

everywhere even as a weak derivative.

Remark 9.12 (Lipschitz perturbation of accretive mappings). The calculation(9.36) applies for a general case if A = A1+A2 with A1 : dom(A) → X m-accretiveand A2 : X → X Lipschitz continuous. Then Aλ := A+λI is m-accretive for λ ≥ ,and then Theorems 9.5, 9.7, and 9.9 obviously extend to this case.

9.3 Excursion to nonlinear semigroups

The concept of evolution governed by autonomous (=time independent) accretivemappings is intimately related to the theory of semigroups, which is an extensively

14This can be seen from the estimate (9.3) for f1 = f2. If X∗ is not uniformly convex, we canuse (9.3) obtained in the limit for those integral solutions which arose by the regularization/time-discretization procedure in the proof of Theorem 9.9, even without having formally proved theiruniqueness in this case.

9.3. Excursion to nonlinear semigroups 315

developed area with lots of nice results, although its application is often ratherlimited. Here we only present a very minimal excursion into it.

A one-parametric collection {St}t≥0 of mappings St : X → X is called aC0-semigroup if St ◦ Ss = St+s, S0 = I, and (t, u) �→ St(u) : [0,+∞)×X → X isseparately continuous15. If, moreover, ‖St(u)− St(v)‖ ≤ eλt‖u− v‖, then {St}t≥0

is called a C0-semigroup of the type λ. In particular, if λ = 0, we speak about anon-expansive C0-semigroup.16

A natural temptation is to describe behaviour of St for all t > 0 by someobject (called a generator) known at the origin t = 0. This idea reflects an ever-lasting ambition of mankind to forecast the future from the present. For this, wedefine the so-called (weak) generator Aw as

Aw(u) = w-limt↘0

St(u)− u

t(9.37)

with dom(Aw) = {u ∈ X, the limit in (9.37) exists}. The relation of non-expansive semigroups with accretive mappings is very intimate:

Proposition 9.13. If {St}t≥0 is a non-expansive semigroup, then Aw is dissipative.

Proof. Take u, v ∈ X and an (even arbitrary) element j∗∈J(u−v). Then⟨j∗,

St(u)− u

t− St(v)− v

t

⟩=

1

t

(⟨j∗, St(u)− St(v)

⟩− ∥∥u− v∥∥2)

≤ 1

t

(∥∥St(u)− St(v)∥∥ − ∥∥u− v

∥∥)∥∥u− v∥∥ ≤ 0 (9.38)

provided St is non-expansive. Considering u, v ∈ dom(Aw), we can pass to thelimit to obtain 〈j∗, Aw(u)−Aw(v)〉 ≤ 0. �

The relation between the semigroup and its generators substantially dependson qualification of X . In general, there even exist non-expansive C0-semigroupspossessing no generator, i.e. dom(Aw) = ∅; such an example is due to Crandalland Liggett [107].

Using (and expanding) our previous results, we obtain a way to generate aC0-semigroup by means of an m-dissipative generator.

Proposition 9.14. Let X and X∗ be uniformly convex and A : dom(A) → X bem-accretive with dom(A) dense in X, and let St(u0) := u(t) with u ∈ C([0, t];X)being a unique integral solution to the problem d

dtu + A(u) = 0 with the initialcondition u(0) = u0. Then:

15This means that both t �→ St(u) and St(·) are continuous. Equivalently, continuity of t �→St(u) is guaranteed by limt↘0 St = I pointwise.

16In literature, from historical reasons, such semigroups are also called, not completely cor-rectly, semigroups of contractions (as if λ were negative).


(i) {St}t≥0 is a non-expansive C0-semigroup whose generator is −A.

(ii) The mapping t �→ A(u(t)) is weakly continuous provided u0 ∈ dom(A).

(iii) Then also the weak derivative ddtu(t) exists for all t≥0 and d

dtu(t)+A(u(t))=0.

Proof. It is obvious that {St}t≥0 is a C0-semigroup. The non-expansiveness ofSt follows from (9.3), cf. Corollary 9.8, i.e. here ‖St(u01) − St(u02)‖ := ‖u1(t) −u2(t)‖ ≤ ‖u01 − u02‖.

As {A(u(t))}t≥0 is bounded in X and, by Milman-Pettis’ theorem, X is re-flexive, we can assume that A(u(tk)) ⇀ ξ inX for (a suitable sequence) tk → t witht ≥ 0. Simultaneously, u(tk) → u(t). As A is m-accretive (hence also maximallyaccretive) and X∗ is uniformly convex, A has a (norm×weak)-closed graph (seethe proof of Theorem 9.5), and thus A(u(t)) = ξ and w-lims→t A(u(s)) = A(u(t)),proving thus (ii).

Then, passing to the limit in 1εu(t+ε)− 1

εu(t) = − 1ε

∫ t+ε

tA(u(s)) ds gives

ddtu(t) = w- limε→0 − 1

ε

∫ t+ε

t A(u(s)) ds = −A(u(t)), proving thus (iii).

For u0∈dom(A), we thus have limt↘01εSt(u0)−1

εu0 = −A(u0). But simulta-neously, by definition (9.37), it is equal to Aw(u0). Thus dom(A) ⊂ dom(−Aw). ByProposition 9.13 and Exercise 3.37, −Aw is accretive and A is maximally accretive,hence dom(A) = dom(−Aw). �

The above assertion can be generalized for Aλ := A + λI m-accretive andthen {St}t≥0 a C0-semigroup of type λ. Also, it conversely holds that a non-expansive C0-semigroup on X , uniformly convex together with its dual, yieldsu(·) : t �→ St(u0) weakly differentiable everywhere, Aw(u(·)) weakly continuous,and d

dtu(t) = Aw(u(t)) for all t ≥ 0;17 let us remark that its generator Aw, whichis dissipative by Proposition 9.13, need not be m-dissipative.

For X∗ uniformly convex and A m-accretive, we have, in fact, proved18 thatthe so-called Crandall-Liggett formula [107]

St(u) = limk→∞

[I+

t

kA

]−k

(u) (9.39)

generates a non-expansive C0-semigroup. In fact, by sophisticated combinatorialarguments, it can be proved for a general Banach space X that the limit in (9.39)is even uniform with respect to t ranging over bounded intervals I.

17If X is a Hilbert space, any maximally accretive mapping is m-accretive, and thus we canprove it simply by taking a maximally accretive extension A of −Aw (which does exists by astandard Zorn-lemma argument) and by applying Proposition 9.14 when realizing that u(t) =St(u0) for any u0 ∈ dom(Aw) ⊂ dom(−A). In a general uniformly convex Banach space we refer,e.g., to Barbu [37, Theorem III.1.2].

18Realize that [I + tkA]−k(u0) = uk

τ with ukτ from (8.5) with fk

τ ≡ 0 and τ = t/k, and then

the convergence limk→∞,τ→0,kτ=t ukτ = limτ→0 uτ (t) = u(t) = St(u0) has been obtained in

the proof of Theorem 9.7 provided u0 ∈ dom(A) and k’s forming an ever-refining sequence ofpartitions of [0, t], while for a general u0 ∈ X this proof must be modified so that the convergenceu0τ → u0 is employed first. The (even Lipschitz) continuity of St(·) follows from the estimate(9.3).

9.3. Excursion to nonlinear semigroups 317

In the Hilbert case, we can use, in particular, Example 9.11. In this situation,the equation d

dtu+A1(u)+A2(u) = 0 induces a C0-semigroup on H of type λ withλ referring to the Lipschitz constant of A2. Again f ∈ L1(I;H) is allowed similarlyas in Theorem 8.13, which completes the previous results.19 In a general Hilbertcase, the relation between non-expansive C0-semigroups and their generators isvery intimate: the generator is always densely defined, and there is a one-to-onecorrespondence between m-accretive mappings and generators of non-expansiveC0-semigroups.

For the linear operator A (and again a Banach space X), the consequence ofProposition 9.14 is the following:

Corollary 9.15. Let X and X∗ be uniformly convex, A0 : dom(A0) → X, withdom(A0) dense in X, be linear, let 〈A0(v), v〉s ≤ 0 and let A0 − I be surjective.Then A0 generates a non-expansive C0-semigroup.

Proof. We take A = −A0. Then A is m-accretive and St(v) := u(t) with u beingthe unique integral solution to d

dtu+A(u(t)) = 0, u(0) = v. �

In fact, the above assertion holds for a general Banach space (even also as aconverse implication), which is known as the Lumer-Phillips theorem [266].

We will still consider a special “semilinear” (but partly non-autonomous)situation, namely that A(t, v) := A1v+A2(t, v) with −A1 being a linear generatorof a non-expansive C0-semigroup {St}t≥0 ⊂ L (X,X) and A2 : I × X → X aCaratheodory mapping qualified later. We call u ∈ C(I;X) a mild solution to theCauchy problem (8.1) if the following Volterra-type integral equation

u(t) = Stu0 +

∫ t

0

St−s

(f(s)−A2(s, u(s))

)ds (9.40)

holds for any t ∈ I. Existence and uniqueness of a mild solution can be shownquite easily:

Proposition 9.16 (Existence and uniqueness). Let A(t, v) := A1v + A2(t, v)with −A1 a linear generator of a non-expansive C0-semigroup {St}t≥0, and theCaratheodory mapping A2 : I × X → X satisfy A2(·, 0) ∈ L1(0, T ;X) and‖A2(t, v1)−A2(t, v2)‖ ≤ (t)‖v1−v2‖ for some ∈L1(0, T ) and v1, v2∈X, and letf ∈L1(I;X) and u0∈X. Then there is just one mild solution u∈C(I;X) to (8.1).

Proof. Uniqueness follows simply by subtracting (9.40) written for two solu-

tions u1 and u2, which gives u12(t) := u1(t) − u2(t) =∫ t

0 St−s(A2(s, u2(s)) −19In fact, one can still prove that, if A1(u0) ∈ H, then there is u ∈ W 1,∞(I;H) a strong

solution to ddt

u + A1(u) + A2(u) = f , u(0) = u0, and even d+

dtu + A1(u) + A2(u) = f holds

everywhere on I where d+

dtdenotes the right derivative, cf. e.g. [37, Theoem III.2.5.].


A2(s, u1(s))) ds, hence

∥∥u12(t)∥∥ ≤

∫ t

0

∥∥St−s

∥∥L (X,X)

∥∥A2(s, u2(s))−A2(s, u1(s))∥∥ds

≤∫ t

0

(s)∥∥u12(s)

∥∥ ds (9.41)

because St−s is non-expansive, from which u12 = 0 follows by Gronwall’s inequal-ity.

For the existence we use Banach’s fixed point Theorem 1.12. Consideringu1, u2 ∈ C(I;X), we calculate u1, u2 ∈ C(I;X) as ui(t) := Stu0 +

∫ t

0 St−s

(f(s)−

A2(s, ui(s)))ds, i = 1, 2. Like (9.41), we can estimate ‖u1(t) − u2(t)‖ ≤∫ t

0 (t)‖u1(s)−u2(s)‖ds so that ‖u1−u2‖C(0,t1;X) ≤( ∫ t1

0 (s)ds)‖u1−u2‖C(0,t1;X).

Hence the mapping u �→ u is a contraction on C(0, t1;X) if t1 > 0 is so small that∫ t10

(s) ds < 1, and has thus a fixed point u ∈ C(0, t1;X), being obviously a mildsolution on [0, t1]. Then, starting from u(t1) instead of u0, we get a mild solution

on [t1, t2] with t2 > 0 so small that∫ t2t1

(s) ds < 1. Altogether, for t ∈ [t1, t2], itholds that

u(t) = St−t1u(t1) +

∫ t

t1

St−s

(f(s)−A2(s, u(s))

)ds

= St−t1

(St1u0 +

∫ t1

0

St1−s

(f(s)−A2(s, u(s))

)ds)+

∫ t

t1

St−s

(f(s)−A2(s, u(s))

)ds

= Stu0 +

∫ t

0

St−s

(f(s)− A2(s, u(s))

)ds,

so we have obtained a mild solution on [0, t2]. As ∈ L1(0, T ), we can continuesuch prolongation until some tk ≥ T . �

Proposition 9.17. Let A, f and u0 be qualified as in Proposition 9.16. Then:

(i) (Consistency.) Any strong solution to ddtu+A1u+A2(t, u) = f , u(0) = u0,

with −A1 a generator of a linear C0-semigroup {St}t≥0 is a mild solution.

(ii) (Selectivity I.) If the mild solution is weakly differentiable, then it is thestrong solution, too.

(iii) (Selectivity II.) If X∗ is uniformly convex, f ∈W 1,1(I;X), u0∈dom(A1),and A2 time independent, then the mild solution is also the strong solution.

Proof. Obviously, 1εSt+εv − 1

εStv = (1εSε − 1εI)Stv = St(

1εSε − 1

εI)v, hence in the

limit ddtStv = AwStv = StAwv = −A1Stv, where

ddt denotes the weak derivative

and Aw = −A1 is the generator of {St}t≥0.Then, as to (i), it holds that d

dsSt−su(s) = A1St−su(s) + St−sddsu(s) =

A1St−su(s)+St−s(−A1u(s)+f(s)−A2(s, u(s))) = St−s(f(s)−A2(s, u(s))), whichgives (9.40) after the integration over [0, t].

9.4. Applications to initial-boundary-value problems 319

As to (ii), differentiating (9.40), one obtains

du

dt=

dSt

dtu0 +

∫ t

0

dSt−s

dt

(f(s)−A2(s, u(s))

)ds+ f(t)−A2(t, u(t))

= AwStu0 +

∫ t

0

AwSt−s

(f(s)−A2(s, u(s))

)ds+ f(t)−A2(t, u(t))

= Awu(t) + f(t)−A2(t, u(t)).

As to (iii), by Theorem 9.5, our problem possesses a strong solution u. Bythis Proposition 9.17(i), u is also a mild solution. By Proposition 9.16, there is noother mild solution. �Remark 9.18 (Non-autonomous systems20). For the general time-dependent A asused in (8.1), instead of a one-parametric C0-semigroup of type λ, it is naturalto consider a two-parametric collection of mappings {Ut,s : X → X}0≤s≤t suchthat Ut,t = I, Ut,ϑ ◦ Uϑ,s = Ut,s for any 0 ≤ s ≤ ϑ ≤ t, and ‖Ut,s(u)− Ut,s(v)‖ ≤eλ(t−s)‖u − v‖ and (t, s) �→ Ut,s(v) is continuous for any u, v ∈ X . If I + λA(t, ·)are accretive for all t ≥ 0, {Ut,s}0≤s≤t can be generated by the Crandall-Liggettformula (9.39) naturally generalized as

Ut,s(u) := limk→∞

k∏i=1

[I+

t−s

kA(s+ i

t−s

k, · )]−1

(u). (9.42)

For the autonomous case when A(t, ·) ≡ A, we can put simply St := U0,t to obtainthe previous situation; note that then also Ut,s ≡ St−s and (9.42) just coincideswith (9.39).

9.4 Applications to initial-boundary-value problems

Example 9.19 (Nonlinear heat transfer). We consider heat transfer in a homoge-neous isotropic but temperature-dependent medium (8.197) moving by a forcedadvection with the given velocity field �v:

c(θ)(∂θ∂t

+ �v · ∇θ)− div

(κ(θ)∇θ

)+ c0(x, θ) = g. (9.43)

We consider the time-independent Neumann boundary condition κ(θ) ∂θ∂ν = h andthe initial condition θ|t=0 = θ0. We first apply the enthalpy transformation andalso the Kirchhoff transformation (8.198), which turns the outlined problem intothe form

∂u

∂t+ �v · ∇u−Δβ(u) + γ(x, u) = g in Q,

∂

∂νβ(u) = h on Σ,

u(0, ·) = u0 on Ω,

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(9.44)

20First relevant papers are by Browder [75], Crandall, Pazy [110], and Kato [225].


with β(r) := [ κ ◦ c−1](r), cf. Example 8.71, and γ(x, r) := c0(x, c−1(r)) and

u0 ∈ c (θ0). The m-accretive mapping approach, cf. also Remark 3.25, requires�v∈W 1,∞(Ω;Rn) such that div�v ≤ 0 and (�v|Σ) · ν = 0 and then can be based onthe setting:

X := L1(Ω), A(u) := �v · ∇u −Δβ(u) + γ(x, u), (9.45a)

dom(A) :={u∈L1(Ω); Δβ(u)− �v · ∇u∈L1(Ω),

∂

∂νβ(u) = h

}; (9.45b)

of course, Δβ(u) − �v · ∇u =: f is meant in the sense of distributions, i.e. 〈f, z〉 =β(u)Δz + u(�v ·∇z)− u (div�v) z for any z∈D(Ω). If γ(x, ·) is Lipschitz continuouswith the Lipschitz constant , then Aλ is accretive for λ≥ , which follows from∫Ω (γ(u)− γ(v) + λ(u−v)) sign(u−v) dx ≥ ∫

Ω(− |u−v| + λ|u−v|) dx ≥ 0. ThenTheorem 9.9 provides existence of an integral solution for physically natural dataqualification, i.e.

g ∈ L1(I;L1(Ω)) ∼= L1(Q) and w0 ∈ L1(Ω), (9.46)

so that the heat sources have a finite energy (without any further restrictions),

while for h we required h ∈ L2#′(Γ) in Proposition 3.22.21

In fact, we need β = κ ◦ c−1 to be Lipschitz continuous and increasing, andwe do not need c to be strictly increasing, and thus c(·) ≥ ε > 0 need not be upper-bounded. Even more, c can be a monotone set-valued mapping which correspondsto Dirac distributions in c. This is an enthalpy formulation of the Stefan problem.Then u0 ∈ c (θ0) is to be determined because the initial temperature θ0 need notbear enough information if c jumps at θ0; each such a jump is related to therespective latent heat of the particular phase transformations.

Example 9.20 (Scalar conservation law22). In view of Section 3.2.4, we considerthe initial-boundary-value problem

∂u

∂t+

∂

∂xF (u) = g in Q := (0, T )× (0, 1),

u(·, 0) = uD on (0, T ),

u(0, ·) = u0 on Ω:= (0, 1).

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (9.47)

The m-accretive mapping approach requires F strongly monotone, cf. Proposi-tion 3.26, and then we obtain an integral solution u ∈ C(I;L1(0, 1)) if g ∈ L1(Q)and u0 ∈ L1(0, 1). A special case F (r) = 1

2r2 leads to the so-called Burgers equa-

tion

∂u

∂t+ u

∂u

∂x= g in Q := (0, T )× (0, 1), u|x=0 = uD, u|t=0 = u0. (9.48)

21For a general time-dependent boundary heat flux h ∈ L1(Σ) see [361] and also Sections 12.1and 12.7–12.9 below.

22See Zeidler [427, Vol.III, Sect.57.6].


The above theory, however, does not apply directly since F is now not stronglymonotone. Assuming u0 ≥ 0, uD ≥ 0, we can expect u ≥ 0, cf. also Exercise 8.87,and then modify F (r) = 1

2 |r|r which is strictly (but not strongly) monotone.

Then we modify dom(A) from (3.39b) for {u∈ L∞(0, 1); u(0) = uD,ddx(|u|u) ∈

L1(0, 1) in the weak sense}, without requiring u ∈ W 1,1(0, 1).

Example 9.21 (Conservation law on Rn). 23 In view of Remark 3.27, we can con-sider also the initial-value problem

∂u

∂t+

n∑i=1

∂

∂xiFi(u) = g in Q := (0, T )× Rn,

u(0, ·) = u0 on Rn.

⎫⎪⎪⎬⎪⎪⎭ (9.49)

Assuming F ∈ C1(R;Rn) and lim sup|u|→0 |F (u)|/|u| < ∞, the mapping A defined

as the closure in L1(Rn) × L1(Rn) of the mapping u �→ div(F (u)) : C10 (R

n) →C0(Rn) is m-accretive on L1(Ω), and then we obtain an integral solution u ∈C(I;L1(Rn)) if g ∈ L1(Q) and u0 ∈ L1(Rn).

Example 9.22 (Hamilton-Jacobi equation). In view of Remark 3.28, the (one-dimensional) Hamilton-Jacobi equation

∂u

∂t+ F

(∂u∂x

)= g in Q := (0, T )× (0, 1),

u|Σ = 0 on Σ := (0, T )× {0, 1},u(0, ·) = u0 on Ω := (0, 1),

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (9.50)

with F :R→R increasing, has an integral solution u ∈ C(Q) if g ∈ L1(I;C([0, 1]))and u0 ∈ C([0, 1]).

Example 9.23 (Nonlinear test I). Consider again the quasilinear boundary-valueproblem (8.167). Being inspired by Section 3.2.2 (i.e. accretivity of Δp in Lq(Ω)and the concrete form of the duality mapping J in Lq(Ω), see Propositions 3.16and 3.13), we can test (8.167) by |u|q−2u, q ≥ 1, as we did in (9.9).24 The particularterms can be estimated as∫

Ω

∂u

∂t|u|q−2u dx =

1

q

∫Ω

∂|u|q∂t

dx =1

q

d

dt‖u‖qLq(Ω), (9.51a)∫

Ω


)|u|q−2u dx

=

∫Ω

|∇u|p−2∇u · ∇(|u|q−2u)dx−

∫Γ

|∇u|p−2 ∂u

∂ν|u|q−2u dS

23See Barbu [38, Sections 2.3.2 and 4.3.4], Dafermos [114, Chap.VI], or Miyadera [287, Chap.7].Other techniques are exposed in Malek at al. [268, Chap.2].

24The calculations (9.51) are only formal unless we have proved regularity of u in advance. Inthe context of results we have proved, a rigorous derivation is to be made by time discretization.


= (q−1)

∫Ω

|∇u|p|u|q−2 dx+

∫Γ

|u|q2+q−2 − h|u|q−2u dS, (9.51b)∫Ω

|u|q1−2u|u|q−2u dx = ‖u‖q1+q−2Lq1+q−2(Ω)

, (9.51c)

cf. Section 3.2.2 for (9.51b). Altogether, abbreviating pi = qi+q−2 for i = 1, 2 andrealizing that |∇u|p|u|q−2 = (p/(p+q−2))p|∇|u|(p+q−2)/p|p, we get

1

q

d

dt

∥∥u∥∥qLq(Ω)

+(q−1) pp

(p+q−2)p∥∥∇|u|(p+q−2)/p

∥∥pLp(Ω)

+∥∥u∥∥p1

Lp1(Ω)+∥∥u∥∥p2

Lp2(Γ)

=

∫Ω

g|u|q−2u dx+

∫Γ

h|u|q−2u dS

≤ ∥∥g∥∥Lq(Ω)

∥∥ |u|q−1∥∥Lq′ (Ω)

+∥∥h∥∥

Lp2/(q2−1)(Γ)

∥∥ |u|q−1∥∥Lp2/(q−1)(Γ)

≤ ∥∥g∥∥Lq(Ω)

(1+∥∥u∥∥q

Lq(Ω)

)+

q2−1

p2

∥∥h∥∥p2/(q2−1)

Lp2/(q2−1)(Γ)+

q−1

p2

∥∥u∥∥p2

Lp2(Γ). (9.52)

We assume g ∈ L1(I;Lq(Ω)), h ∈ Lp2/(q2−1)(Σ), and u0 ∈ Lq(Ω), and use Gron-wall’s inequality to get the estimate of u in L∞(I;Lq(Ω)) ∩ Lq1+q−1(Q) and of|u|(p+q−2)/p in Lp(I;W 1,p(Ω)). If p = 2, from the term (q−1)|u|q−2|∇u|2 in (9.51b)we obtain through (1.46) still an estimate of u in Lq(I;W 2/q−ε,q(Ω)) if q ≥ 2.

Having the estimate∫Q(q−1)|∇u|p|u|q−2dxdt ≤ C1 and supt ‖u(t, ·)‖Lq(Ω) ≤

C2, we can derive the estimate of ∇u in a suitable Lebesgue space even for anarbitrary p > 1. Take r ≥ 1. By Holder inequality, assuming r < p,∫

Q

|∇u|rdxdt =∫Q

|∇u|r|u|(q−2)r/p|u|(2−q)r/pdxdt

≤(∫

Q

|∇u|p|u|q−2dxdt)r/p( ∫

Q

|u|(2−q)r/(p−r)dxdt)(p−r)/p

=( C1

q − 1

)r/p(∫ T

0

∥∥u(t, ·)∥∥(2−q)r/(p−r)

L(2−q)r/(p−r)(Ω)dt)(p−r)/p

. (9.53)

By the Gagliardo-Nirenberg inequality (1.39) with the equivalent norm onW 1,r(Ω)chosen as ‖u‖Lq(Ω) + ‖∇u‖Lr(Ω;Rn), one obtains:

∥∥u(t, ·)∥∥L(2−q)r/(p−r)(Ω)

≤ CGN

(∥∥u(t, ·)∥∥Lq(Ω)

+∥∥∇u(t, ·)∥∥

Lr(Ω;Rn)

)λ∥∥u(t, ·)∥∥1−λ

Lq(Ω)

≤ CGNC1−λ2

(C2 +

∥∥∇u(t, ·)∥∥Lr(Ω;Rn)

)λ(9.54)

for

p− r

(2− q)r≥ λ

(1r− 1

n

)+ (1− λ)

1

q. (9.55)


We raise (9.54) to the power (2−q)r/(p−r) power, use it in (9.53), and chooseλ := (p−r)/(2−q):( ∫ T

0

‖u(t, ·)‖(2−q)r/(p−r)

L(2−q)r/(p−r)(Ω)dt)(p−r)/p

≤( ∫ T

0

Cp−r2−qGN C

(1−λ)(p−r)2−q

2

(C2 + ‖∇u(t, ·)‖Lr(Ω;Rn)

)λ(2−q)rp−r dt

) p−rp

≤( ∫ T

0

Cp−r2−qGN C

1−λ(p−r)2−q

2 2r−1(Cr

2 + ‖∇u(t, ·)‖rLr(Ω;Rn)

)dt) p−r

p

= C3 + C4

(∫Q

|∇u|rdxdt)(p−r)/p

(9.56)

for suitable C3 and C4. Substituting this choice of λ := (p− r)/(2− q) into (9.55),one gets, after some algebra,

r ≤ qp+ np+ qn− 2n

q + n, and also 0 ≤ p− r

2− q≤ 1 and r < p. (9.57)

Since always (p− r)/p < 1, by linking (9.53) with (9.56), we get the bound of ∇uin Lr(Q;Rn).

Example 9.24 (Nonlinear test II). Considering again the problem (8.167), we candifferentiate it in time and test it by | ∂∂tu|q−2 ∂

∂tu, q ≥ 1, as we did in (9.11).25

The particular terms arising on the left-hand side can be treated as∫Ω

∂2u

∂t2

∣∣∣∂u∂t

∣∣∣q−2 ∂u

∂tdx =

1

q

d

dt

∫Ω

∣∣∣∂u∂t

∣∣∣q dx =1

q

d

dt

∥∥∥∂u∂t

∥∥∥qLq(Ω)

, (9.58a)∫Ω

− ∂

∂tdiv(|∇u|p−2∇u

)∣∣∣∂u∂t

∣∣∣q−2 ∂u

∂tdx =

∫Ω

∂

∂t

(|∇u|p−2∇u

)·∇(∣∣∣∂u

∂t

∣∣∣q−2 ∂u

∂t

)dx−

∫Γ

∂

∂t

(|∇u|p−2 ∂u

∂ν

)∣∣∣∂u∂t

∣∣∣q−2 ∂u

∂tdS

≥∫Ω

(q−1)8p−8

p2

(∂|∇u|p/2∂t

)2∣∣∣∂u∂t

∣∣∣q−2

dx

+

∫Γ

(q2−1)|u|q2−2∣∣∣∂u∂t

∣∣∣q − ∂h

∂t

∣∣∣∂u∂t

∣∣∣q−2 ∂u

∂tdS, (9.58b)∫

Ω

∂(|u|q1−1u)

∂t

∣∣∣∂u∂t

∣∣∣q−2 ∂u

∂tdx =

∫Ω

(q1 − 1)|u|q1−2∣∣∣∂u∂t

∣∣∣q dx ≥ 0 (9.58c)

cf. the calculations in (8.174). Like (8.174), it requires p > 1. Assuming h constantin time so that ∂

∂th simply vanishes, we obtain

1

q

d

dt

∥∥∥∂u∂t

∥∥∥qLq(Ω)

≤∫Ω

∂g

∂t

∣∣∣∂u∂t

∣∣∣q−2 ∂u

∂tdx ≤

∥∥∥∂g∂t

∥∥∥Lq(Ω)

(1 +

∥∥∥∂u∂t

∥∥∥qLq(Ω)

). (9.59)

25Again, the calculations in this example are only formal unless we have proved regularity ofu in advance. A rigorous derivation would have to be made by time discretization.


By Gronwall’s inequality, we get u ∈ W 1,∞(I;Lq(Ω)) provided g ∈ W 1,1(I;Lq(Ω))and div(|∇u0|p−2∇u0)− |u0|q1−2u0 = ∂

∂tu|t=0 ∈ Lq(Ω).

If p = 2, we obtain a term (q−1)| ∂∂tu|q−2| ∂∂t∇u|2 in (9.58b), cf. (8.174). Then,

from (1.46), we obtain still an estimate of ∂∂tu in Lq(I;W 2/q−ε,q(Ω)) if q ≥ 2.

Moreover, if q1=2, the respective term, i.e. now∫Ω| ∂∂tu|qdx ≥ 0, gives also

the estimate W 1,q(I;Lq(Ω)). This estimate is weaker in comparison with the usualW 1,∞(I;Lq(Ω))-estimate (9.7) but, contrary to it, is uniform with respect to q →∞ if g∈W 1,∞(I;L∞(Ω)), u0∈L∞(Ω), and Δpu0∈L∞(Ω). Thus we get a uniformestimate L∞(I;Lq(Ω)), from which the boundedness of u in L∞(Q) follows. Theidea of passing to L∞-bounds by increasing q is called Moser’s trick [295] but itis usually organized in a much more sophisticated way.

Remark 9.25 (Nonlinear test I revisited: case q = 1). Note that the Lr-estimateof ∇u obtained in Example 9.23 does not work for q = 1 because of the factor1/(q−1) in (9.53). Boccardo and Gallouet [56, Formula (2.7)] give the bound for∫Q|∇u|p/(1+|u|)1+ε0 dxdt, which then gives, essentially by the same procedure as

above, the exponent r < (p+np−n)/(n+1). In fact, instead of the test by |u|q−2ufor q = 1, one can rather use the test by signε(u) from (3.23) like in (3.2.2) orby χ(u) := 1 − 1

(1+u)ε , ε > 0, proposed by Feireisl and Malek [145]. For p = 2,

cf. (12.17) below.

qualification of quality

g h u0 p q of u

L1(I;Lq(Ω)) Lp2

q2−1 (Σ) Lq(Ω) > 1 ≥ 1 L∞(I;Lq(Ω)) ∩ Lp1(Q)

= 2 ≥ 2 Lq(I;W 2/q−ε,q(Ω))

W 1,1(I;Lq(Ω)) constant Δpu0∈Lq(Ω) > 1 ≥ 1 W 1,∞(I;Lq(Ω))

in time u0∈Lq1q−q(Ω) = 2 ≥ 2 W 1,q(I;W 2/q−ε,q(Ω))

Table 5. Summary of Examples 9.23–9.24; p1 := q1+q−2 and p2 := q2+q−2.

Exercise 9.26 (Heat equation with advection: a nonlinear test). Consider againthe heat equation in the enthalpy formulation ∂

∂tu + v · ∇u − Δβ(u)+γ(u) = g,cf. (9.44), and assume div v ≤ 0 and v|Γ · ν ≥ 0 as in Exercise 2.91, and test itby |u|q−2u, q ≥ 1, to obtain the estimate of u in L∞(I;Lq(Ω)). For β stronglymonotone and q ≥ 2, from (1.46) derive also an estimate in Lq(I;W 2/q−ε,q(Ω)).26

26Hint: Use Example 9.23 but modify (9.51b) to∫ΩΔβ(u)|u|q−2udx = (1−q)

∫Ωβ′(u)|u|q−2|∇u|2dx+

∫Γ(h− b(u))|u|q−2udS ≤

∫Γh|u|q−2u dS

because β′(r) ≥ 0 and b(r)r ≥ 0 in (8.199). Moreover, treat the advective term as in (3.31).


Exercise 9.27 (Conservation law regularized). For ε > 0 fixed, as in Exercise 8.87,consider again

∂u

∂t+ div

(F (u)

)− εΔu = g, u|t=0 = u0, u|Σ = 0, (9.60)

test it by |u|q−2u, q ≥ 1, and prove an a-priori estimate in L∞(I;Lq(Ω)) and, ifq ≥ 2, also in Lq(I;W 2/q−ε,q(Ω)).27

Remark 9.28 (Positivity of temperature). The estimates in Exercise 9.26 can beinterpolated as in Example 9.23 with Remark 9.25 to obtain the estimate of ∇uin Lr(Q;Rn) for 1 ≤ r < n+2

n+1 . Often, the meaning of the solution u to (9.44) isthe absolute temperature and then the desired property would be its positivity.Indeed, such positivity can be proved by a comparison argument similarly likein [146, Sect. 4.2.1]. Assuming inf u0 > 0, h ≥ 0, and γ(u) − g ≤ C|u|ω−2ufor some ω ≥ 2 and C ≥ 0, we can compare u with some v spatially constant,i.e. v(t, x) = y(t), such that y solves the initial-value problem for the ordinarydifferential equations dy

dt + |y|ω−2y = 0 with y(0) = y0 := inf u0 > 0. This problemhas just one solution which is positive.28 Then v solves the initial-boundary-valueproblem

∂v

∂t+ �v · ∇v −Δβ(v) + C|v|ω−2v = 0 in Q,

∂

∂νβ(v) = 0 on Σ,

u(0, ·) = y0 := inf u0 on Ω.

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(9.61)

The comparison between (9.44) and (9.61) can formally29 be performed by sub-tracting the respective weak formulations and testing them by (u−v)−. Important

27Hint: Denote by Tq : R→ R the inverse mapping to r �→ |r|q−2r, i.e. Tq(r) = |r|(2−q)/(q−1)r,and by Fi,q : R→ R the primitive function to Fi ◦Tq : R→ R such that Fi,q(0) = 0, i = 1, . . . , n.Then the F -term, if tested as suggested, vanishes; indeed, by using Green’s formula twice,∫

Ωdiv(F (u))|u|q−2u dx = −

∫ΩF (u) · ∇(|u|q−2u) dx = −

∫Ω[F ◦ Tq ](v) · ∇v dx

= −∫Ω

n∑i=1

[Fi ◦ Tq](v)∂v

∂xidx = −

∫Ω

n∑i=1

∂Fi,q(v)

∂xidx = −

∫Ωdiv(F1,q , . . . , Fn,q)(v) dx

=

∫Ω(F1,q , . . . , Fn,q)(v) · (∇ 1) dx−

∫Γ

n∑i=1

Fi,q(v)νi dS = −∫Γ

n∑i=1

Fi,q(v)νi dS = 0,

where we used the substitution v := |u|q−2u = T−1q (u). The remaining terms have the pos-

itive sign as in Example 9.23 for p = 2. If q ≥ 2, use (1.46) to get the fractional-derivativeestimate. Note that the growth of F can be superlinear: the condition (2.55a), requiring

NF :L(p∗−ε)′ (Ω)→Lp′(Ω;Rn), implies |F (r)| ≤ C+|r|(p∗−ε)′/p′ .28For ω = 2, one has y(t) = e−Cty0 while, for ω > 2, one can find y(t) = (y0+C(ω−2)t)1/(2−ω).29If the assumptions on the data do not give sufficient integrability for such a test, one has to

make it in a suitable approximation and then refine it to the limit.


phenomena are that the advective term∫Q �v ·∇(u−v)(u−v)−dxdt is non-negative

by the argument as in Exercise 2.91 and similarly the Δβ-term gives(∇β(u)−∇β(v))·∇(u−v)− = ∇β(u)·∇(u−v)−

= χ{u≤v}∇β(u)·∇u = χ{u≤v}β′(u)|∇u|2 ≥ 0 (9.62)

a.e. on Q, which can be shown by a smoothening like in the proof of Propo-sitions 3.16 or 3.22; here χ{u≤v} denotes the characteristic function of the set{(t, x) ∈ Q; u(t, x) ≤ v(t, x)}. As a result, this comparison gives u ≥ v > 0 on Q.

9.5 Applications to some systems

The accretivity technique works, in a limited extent, even for systems of equationson special occasions.

An interesting example30 is the abstract initial-value problem (the Cauchyproblem) for the 2nd-order doubly nonlinear equation

d2u

dt2+ A

(dudt

)+B(u) = f(t), u(0) = u0 ,

du

dt(0) = v0, (9.63)

which can equivalently be written as the system of two 1st-order equations:

du

dt− v = 0, u(0) = u0, (9.64a)

dv

dt+A(v) +B(u) = f, v(0) = v0. (9.64b)

Assumptions we make are the following, cf. also Theorem 11.33(ii) below:

A : V → V ∗ monotone and radially continuous, (9.65a)

B = B1 +B2 with

B1 : V → V ∗ linear, continuous, symmetric, i.e. B∗1 = B1, and

〈B1u, u〉 ≥ c0‖u‖2V − c1‖u‖2H , c0 > 0, c1 ≥ 0,

B2 : H → H Lipschitz continuous ( =the Lipschitz constant), (9.65b)

V and H Hilbert spaces. (9.65c)

30Cf. Barbu [38, Sect.4.3.5] where A is admitted set-valued, describing thus a 2nd-order evo-lution variational inequality. For A = 0 and B = div(A(x)∇u) see e.g. Renardy and Rogers [349,Sect.11.3.2].

9.5. Applications to some systems 327

If A ≡ 0, (9.63) is a semilinear hyperbolic equation. We put

X := V ×H, (9.66a)

dom(C) :={(u, v)∈X ; A(v) +B1u∈H, v∈V

}, and (9.66b)

C(u, v) :=(λu − v , λv +A(v) +B(u)

), where (9.66c)

λ ≥ supu∈Vv∈H

(c1 + )(u, v)

〈B1u, u〉+ c1‖u‖2H + ‖v‖2H, λ >

√c1 + − 1; (9.66d)

note that, due to (9.65b), the denominator in (9.66d) is lower bounded by c0‖u‖2V +‖v‖2H . Then, putting w := (u, v), the system (9.64) can be rewritten as

dw

dt+ C(w) − λw = (0, f) , w(0) = (u0, v0) . (9.67)

We endow X with an inner product⟨(u1, v1), (u2, v2)

⟩X×X

:=⟨B1u1, u2

⟩V ∗×V

+c1

⟨u1, u2

⟩H×H

+⟨v1, v2

⟩H×H

(9.68)

and identify X∗ with X itself. Then the duality mapping J is the identity and onecan show C : dom(C) → X is accretive, i.e. monotone with respect to the product(9.68), cf. Remark 3.10; indeed for any w1, w2 ∈ dom(C) one has⟨

C(w1)−C(w2), J(w12)⟩X×X∗ =

(C(w1)−C(w2), w12

)X×X

=⟨B1(λu12−v12), u12

⟩V ∗×V

+ c1(λu12−v12, u12

)X×X

+(λv12 +A(v1)−A(v2) +B1u12 +B2(u1)−B2(u2), v12

)X×X

=⟨A(v1)−A(v2), v12

⟩V ∗×V

+(B2(u1)−B2(u2), v12

)X×X

− c1(v12, u12

)X×X

+ λ(⟨B1u12, u12

⟩V ∗×V

+ c1∥∥u12

∥∥2H+∥∥v12∥∥2H) ≥ 0 (9.69)

where we again abbreviated w12 := w1 −w2, u12 := u1−u2 and v12 := v1− v2, anduse both monotonicity of A and that λ is large enough, cf. (9.66d). Moreover, Cis maximal monotone, which means that for any (f0, f1) ∈ V ×H =: X there isw ≡ (u, v) ∈ dom(C) such that w + C(w) = (f0, f1), i.e.

u+ λu− v = f0, (9.70a)

v + λv +A(v) +B1u+B2(u) = f1. (9.70b)

From (9.70a) one gets u = (v+f0)/(1+λ). Putting this into (9.70b) yields

(1+λ)v +B1v

1+λ+A(v) +B2

(v+f01+λ

)= f1 − B1f0

1+λ∈ V ∗. (9.71)

Since, due to (9.66d), we have 1+λ > (c1+ )/(1+λ), the mapping v �→ (1+λ)v +B1v/(1+λ)+B2((v+f0)/(1+λ)) is monotone, coercive, and bounded as a mapping


V → V ∗. By Proposition 2.20, (9.71) has a solution v ∈ V . Then also u =(v + f0)/(1 + λ) ∈ V and thus, by (9.70b), A(v) + B1u = f − (1+λ)v ∈ H sothat altogether (u, v)∈dom(C). Therefore, C is also m-accretive.

Assuming, in addition, that V0 := {u ∈ V ; B1u ∈ H} is dense in V , wehave dom(C) dense in X .31 Then Theorem 9.732 directly implies the existence ofan integral solution (u, v) ∈ C(I;X) to (9.64), i.e. also a solution u ∈ C(I;V ) ∩C1(I;H) to (9.63), provided u0 ∈ V , v0 ∈ H , f ∈ L1(I;H).

Example 9.29. For V := W 1,20 (Ω), H := L2(Ω), a : Rn → Rn monotone, |a(s)| ≤

C(1 + |s|), and c : R → R Lipschitz continuous, the equation

∂2u

∂t2− div a

(∇∂u

∂t

)−Δu+ c(u) = g (9.72)

with the zero Dirichlet boundary conditions in the weak formulation satisfies allthe above requirements; note that obviously V0 = {u∈W 1,2

0 (Ω); Δu∈L2(Ω)} ⊃W 2,2(Ω) ∩W 1,2

0 (Ω) is dense in W 1,20 (Ω).

Example 9.30 (Linearized thermo-visco-elasticity33). More sophisticated usage ofthe transformation (9.63)→(9.64) is for a system that arises by a linearization ofthe full thermo-visco-elasticity system, cf. (12.4)-(12.5) below34:

∂2u

∂t2− μvΔ

∂u

∂t− μΔu+ α∇θ = g, (9.73a)

∂θ

∂t− κΔθ + αθ0div

∂u

∂t= h, (9.73b)

under the initial conditions u = u0,∂∂tu = v0, θ = θ0, where θ0 is a constant, and

some boundary conditions, say u|Σ = uD with uD constant in time and ∂∂ν θ = 0.

Here the following notation is used:u : Q → Rn is an unknown displacement,θ : Q → R is an unknown temperature,μv≥0 (resp. μ>0) a coefficient related to viscosity (resp. elasticity) response,κ > 0 a coefficient expressing heat conductivity, andα a coefficient expressing thermal expansion,

31Indeed, as V is assumed dense in H, we can first approximate any v ∈ H by some v ∈ V .Then, for f ∈ H arbitrary, take u ∈ V such that A(v) +B1u = f ∈ H, hence B1u = f −A(v) ∈V ∗, and taking u ∈ V such that B1u1 = A(v), we have B1z = f ∈ H for z = u + u, hence uranges over V0 − u which is, by the assumption, dense in V .

32Note that, e.g., Theorem 8.30 for evolution via monotone mapping A1 := C with lower-orderterms A2 := 0 and A3 := −λI cannot be used directly because we are now not in the situationthat C : X → X∗ with X compactly embedded into a pivot Hilbert space.

33A special case n = 1 and μ = θ0 = 1 is in Zheng [429, Sect.2.7] and μv = 0 is in Jiang andRacke [217, Sect.7.2].

34For simplicity, we consider here λ = λv = γ = 0, c = � = 1, and α in place of α(3λ+2μ). Thelinearization uses the natural assumption that the temperature varies only very slightly aroundθ0 and the process is very slow so that the contribution of the terms quadratic in the velocity∂∂t

u in (12.5) is only small and can be well neglected.


g, h are mechanical loading and heat sources, respectively.Denoting v := ∂

∂tu/√μ and z = θ/

√θ0μ and dividing (9.73a) by

√μ and (9.73b)

by√θ0μ, the system (9.73) transforms into

∂

∂t

⎛⎝uvz

⎞⎠+

⎛⎝ 0 −√μ 0

−√μΔ −μvΔ α

√θ0∇

0 α√θ0 div −κΔ

⎞⎠⎛⎝uvz

⎞⎠ =

⎛⎝ 0g/

√μ

h/√θ0μ

⎞⎠ . (9.74)

The setting X := W 1,2(Ω;Rn) × L2(Ω;Rn) × L2(Ω) with the inner product de-fined by ((u1, v1, z1), (u2, v2, z2)) :=

∫Ω∇u1:∇u2 + v1·v2 + z1z2 dx and dom(A) :=

{(u, v, z)∈X ; A(u, v, z)∈X, u|Γ = uD, v|Γ = 0, ∂∂ν z = 0} with A defined by the

matrix in (9.74) makes A accretive; indeed,(A(u, v, z), (u, v, z)

)=

∫Ω

(−√

μ∇v : ∇u−√μΔu · v − μvΔv · v

+ α√θ0 ∇z · v + α

√θ0 div(v) z − κΔz z

)dx

= μv

∥∥∇v‖2L2(Ω;Rn×n) + κ∥∥∇z

∥∥2L2(Ω;Rn)

≥ 0. (9.75)

The m-accretivity follows by Lax-Milgram Theorem 2.19. Then, by Theorem 9.7,we obtain a unique integral solution u ∈ C(I;W 1,2(Ω;Rn))∩C1(I;L2(Ω;Rn)) andθ ∈ C(I;L2(Ω)) provided g ∈ L1(I;L2(Ω;Rn)), h ∈ L1(I;L2(Ω)).

Example 9.31 (Generalized standard materials [9, 194, 197]). Other usage of thetransformation (9.63)→(9.64) is for a model of Halphen and Nguen’s [197] gen-eralized standard materials35, i.e. isothermal model of materials with internal pa-rameters z ∈ Rm. At small strains, it is governed by the following system of theequilibrium equation for u and of the evolution inclusion for z:

�∂2u

∂t2− div σ = g, (9.76a)

σ = σv

(∂w∂t

)+ ψ′e(w, z), w = e(u) :=

1

2(∇u)+

1

2∇u, (9.76b)

γ(∂z∂t

)+ �ψ′z(w, z) � 0, (9.76c)

where ψ : Rn×nsym × Rm → R with Rn×n

sym the set of n × n symmetric matrices isquadratic positive definite. To be more specific, let us consider

�ψ(e, z) :=1

2(e − Bz)D(e− Bz) +

1

2zLz (9.77)

with D ∈ Rn×n×n×nsym , B ∈ Rn×n×m, and L ∈ Rm×m

sym . The meaning of the variablesand the constants is:

35This covers many special cases, among them so-called Prandtl-Reuss or Maxwell materials,see Alber [9, Chapter 3] for these and many more cases. For the accretive formulation of thePrandtl-Reuss plasticity or of the plasticity with hardening, see also [384].


u : Q → Rn is the displacement,w : Q → Rn×n

sym the strain (as a function of (t, x)),z : Q → Rm the internal parameters ,e : Rn×n → Rn×n

sym the small-strain tensor, cf. (6.22),σv : Rn×n

sym → Rn×nsym a monotone viscous-stress tensor,

σ : Q → Rn×nsym the total stress tensor, here σ = σv(

∂∂tw) + D(w − Bz),

γ : Rm ⇒ Rm a maximal strictly monotone (possibly set-valued) mapping,36

� > 0 mass density,g : Q → Rn an external force.

We have to specify initial conditions for u, ∂∂tu, and z, and boundary conditions

for u; as to the latter point, let us consider zero Dirichlet conditions for simplicity.Let us assume

σv, γ−1 continuous with at most linear growth, (9.78a)

∃ε > 0 ∀e ∈ Rn×nsym : σv(e):e ≥ ε|e|2, (9.78b)

c(0) � 0. (9.78c)

Note that γ−1 indeed does exist since we assumed γ strictly monotone. Denotingv := ∂

∂tu as in (9.64a), the system (9.76) can be written as the first-order system

in terms of (v, w, z) as ∂∂t (v, w, z) + C(v, w, z) = (g, 0, 0) with C defined by

C(v, w, z) := −( div

(σv(e(v))+D(w−Bz)

)�

, e(v) , γ−1(−BD(w−Bz)−Lz

)).

We set X := L2(Ω;Rn)×L2(Ω;Rn×n)×L2(Ω;Rm) with the norm ‖(v, w, z)‖X =(∫Ω �|v|2 + 2�ψ(e, z) dx)1/2, which makes X a Hilbert space, and dom(C) :=

{(v, w, z) ∈ X ; v ∈ W 1,20 (Ω,Rn), A(v, w, z) ∈ X}. This makes C accretive: in-

deed, as J is the identity, for any (v1, w1, z1), (v2, w2, z2) ∈ dom(C), we have theestimate⟨

C(v1, w1, z1)−C(v1, w1, z1), J(v12, w12, z12)⟩X×X∗

=

∫Ω

−div(σv(e(v1))− σv(e(v2)) + D(w12 − Bz12)

) · v12−(e(v12)− Bξ12

)D(w12 − Bz12)− ξ12Lz12 dx

=

∫Ω

(σv(e(v1))− σv(e(v2))± D(w12 − Bz12)

):∇v12

− ξ12(BD(w12−Bz12) + Lz12

)dx ≥ 0 (9.79)

where “±” indicates the terms that cancel each other and where we abbreviatedξi := γ−1(−BD(wi−Bzi) − Lzi) for i = 1, 2 and, as before, v12 := v1 − v2,w12 := w1 −w2, ξ12 := ξ1 − ξ2, etc. We used also that σv(·) is assumed monotone

36When γ(0) is not a singleton, this allows for modelling activated processes in evolution of z.


and that σ:∇v = σ:e(v) because σ is symmetric. The last term in (9.79) is indeednon-negative as γ(·) is monotone. To prove that C is m-accretive, we show, forany (g, g1, g2) ∈ X , existence of some (v, w, z) ∈ dom(C) such that (v, w, z) +C(v, w, z) = (g, g1, g2). Considering V := W 1,2

0 (Ω;Rn)×L2(Ω;Rn×n)×L2(Ω;Rm)and now C : V → V ∗ in the weak formulation, the existence of (v, w, z) ∈ V followsby Browder-Minty theorem 2.18; the radial continuity of C follows by (9.78a)while its coercivity follows by (9.78b,c) if (9.76) is used for (v2, w2, z2) := (0, 0, 0).Moreover, since also C(v, w, z) = (g − v, g1 − w, g2 − z) ∈ X , we have (v, w, z) ∈dom(C). In particular, Theorem 9.7 then gives us existence of a unique integralsolution to (9.76) provided still g ∈ L1(I;L2(Ω;Rn)), u(0, ·) ∈ W 1,2(Ω;Rn) so thatw(0, ·) ∈ L2(Ω;Rn×n

sym ), v(0, ·) = ∂∂tu(0, ·) ∈ L2(Ω;Rn), and z(0, ·) ∈ L2(Ω;Rm).

For the more difficult non-dissipative case σv = 0 we refer to Alber [9, Chap 4].

Example 9.32 (Phase-field system37). Solidification processes can be described bythe system

∂u

∂t= Δu+ ζ

∂v

∂t+ g, u|t=0 = u0, (9.80a)

ξ∂v

∂t= ξΔv − 1

ξc(v)− u, v|t=0 = v0, (9.80b)

for the unknown u and v having the meaning of a temperature and an orderparameter, respectively, and with fixed ζ > 0 and (small) ξ > 0. Consideringzero Dirichlet boundary conditions, we define X := L2(Ω)2, dom(A) = {z ∈W 1,2

0 (Ω); Δz∈L2(Ω)}2, and A := A1+A2 with A1(u, v) := ( ζξu−Δ(u+ζv),−Δv)

and A2(u, v) := ( ζξ2 c(v),

1ξ2 c(v)+

1ξu). Obviously, ∂

∂t (u, v)+A(u, v) = (g, 0) is just

(9.80), namely (9.80b) multiplied by ζ/ξ is added to (9.80a) and (9.80b) is di-vided by ξ. Considering c : R → R Lipschitz continuous and the Hilbert spaceX endowed with the inner product ((u1, v1), (u2, v2)) :=

∫Ωu1u2 + ζ2v1v2 dx and

identified with its own dual, the linear operator A1 is accretive,

⟨A1(u, v), J(u, v)

⟩X×X∗ =

(A1(u, v), (u, v)

)X×X

=

∫Ω

(ζξu−Δ(u+ζv)

)u

− ζ2Δv v dx =

∫Ω

ζ

ξu2 + |∇u|2 + ζ∇u ·∇v + ζ2|∇v|2 dx

≥ ζ

ξ

∥∥u∥∥2L2(Ω)

+1

2

∥∥∇u∥∥2L2(Ω;Rn)

+ζ2

2

∥∥∇v∥∥2L2(Ω;Rn)

≥ 0, (9.81)

while A2 is Lipschitz continuous. By Lax-Milgram Theorem 2.19, A1 can be shownm-accretive. Besides, dom(A) ⊃ W 2,2(Ω)2 ∩W 1,2

0 (Ω)2 shows dom(A) dense in X .Then, by Theorem 9.7, we obtain a unique integral solution (u, v) ∈ C(I;L2(Ω)2)provided g ∈ L1(I;L2(Ω)) and u0, v0 ∈ L2(Ω).

37For more details see Sect. 12.5 below.


Example 9.33. Modification of Example 9.20 leads naturally to a system of mequations for u = (u1, . . . , um) : (0, T )× (0, 1) → Rm:

∂ui

∂t+

∂

∂xFi(ui) +Gi(u1, . . . , um) = 0 in Q := (0, T )× (0, 1),

ui(·, 0) = 0 on (0, T )× {0},ui(0, ·) = u0 on Ω:= (0, 1), i = 1, . . . ,m.

⎫⎪⎪⎬⎪⎪⎭(9.82)

To apply the m-accretive mapping approach, we assume Fi strongly monotone andG : Rn → Rn Lipschitz continuous, and then put:

X := L1(0, 1;Rm), A(u) :=( ∂

∂xF1(u1), . . . ,

∂

∂xFm(um)

)+G(u), (9.83a)

dom(A) :={u∈W 1,1(0, 1;Rm);

∂

∂xF (u)∈L1(0, 1;Rm), u(0) = 0

}. (9.83b)

Choosing λ greater than the Lipschitz constant of G, the mapping Aλ : A + λIwill be accretive.

Remark 9.34 (Carleman’s system38). Some other systems that give rise to accretivemappings exist, as e.g. the following two-dimensional hyperbolic system

∂u

∂t+

∂u

∂x1+ u2 − v2 = 0, (9.84a)

∂v

∂t+

∂v

∂x2+ v2 − u2 = 0, (9.84b)

considered on Ω := (R+)2 together with the boundary conditions u(t, 0, x2) =v(t, x1, 0) = 0 and the initial conditions u(0, x) = u0 ≥ 0 and v(0, x) = v0 ≥ 0.The setting dom(A) := {(u, v) ∈ L1((R+)2)2 ∩ L∞((R+)2)2; ∂

∂x1u + u2 − v2 ∈

L1((R+)2), ∂∂x2

v + v2 − u2 ∈ L1((R+)2), u, v ≥ 0, u(t, 0, x2) = v(t, x1, 0) = 0}makes the underlying mapping A accretive.


In general, see the monographs mentioned in Sect. 3.4. Some more detailed com-ments are as follows.

The notion of integral solution has been introduced by Benilan and Brezis [48]for X a Hilbert space and then by Benilan [45, 46] for X a general Banach space.See also Barbu [37, Sect.III.2.1], Deimling [118, Sect.14.3], Hu and Papageorgiou[209, Part I, Sect.III.8]. An equivalent definition involving the inequality

e−2λt∥∥u(t)−v

∥∥2≤ e−2λs∥∥u(s)−v

∥∥2+ 2

∫ t

s

e−2λϑ⟨f(ϑ)−A(v), u(ϑ)−v

⟩sdϑ (9.85)

38See Miyadera [287, Examples 2.3, 4.10, 6.2].


has been used by Miyadera [287, Sect.5.1]. Besides, an alternative definition

∥∥u(t)−v∥∥ ≤ ∥∥u(s)−v

∥∥+ ∫ t

s

⟨f(ϑ)−A(v), u(ϑ)−v

⟩++ λ

∥∥u(ϑ)−v∥∥ dϑ (9.86)

with 〈u, v〉+ = infε>01ε‖u+εv‖− 1

ε‖u‖ can be used, see Showalter [383, Chap.IV.8]or Zeidler [427, Chap.57]; it holds that 〈u, J(v)〉 ≤ 〈u, v〉+‖v‖. This definitionmakes some estimates easier, e.g. it shows that (9.3) holds also for integral solu-tions. A combination of (9.85) and (9.86) is in Barbu [38, Sect.4.1.1].

There is an alternative technique to prove existence of an integral solutionto (8.4) based on a regularization of A: instead of the Rothe approximation andTheorem 9.5, it is possible to use the solution of d

dtu + [Yε(A)](u) = f whereYε(A) := ε−1J(I− (I+ εJ−1A)−1) is the Yosida approximation of A; for X = Rn

cf. (2.164b) and for X general see Remark 5.18. For this approach see Barbu[37, Sections 3.1-2] and [38, Sections 4.1.2], Miyadera [287, Chap.3], Yosida [425,Sect.XIV.6-7], or Zeidler [427, Sect.31.1].

Uniqueness in nonreflexive case (not proved here) can be found in Barbu [38,Sections 4.1.1] (by a smoothing method) or Showalter [383, Sect.IV.8] (by Rothe’smethod). This is related to the Crandall-Liggett formula for the general Banachspace, see, e.g. Barbu [37, Sect.III.1.2] or Pavel [326, Sect.3.2].

An accretive approach to the Klein-Gordon equation, cf. Exercise 11.41, is inBarbu [38, Sect. 4.3.5], Cazenave and Haraux [90], or Kobayashi and Oharu [234].

For non-expansive semigroups see Barbu [37], Belleni-Morante and McBridge[40, Chap.5], Cioranescu [95, Chap.VI], Crandall and Pazy [109], Hu and Papa-georgiou [209, Part I, Sect.III.8], Ito and Kappel [212, Chap.5], Miyadera [287,Chap.3-4], Pavel [326, Chap.II], Pazy [330, Chap.6], Renardy and Rogers [349,Chap.11], or Zeidler [427, Sect.57.5]. In case of X being a Hilbert case, see inparticular Barbu [37, Sect.4.1], Brezis [66], Zeidler [427, Sect.31.1] or Zheng [429,Chap.II]. Semilinear parabolic equations treated on the base of the convolution(9.40) and their mild solution are in Cezenave and Haraux [90], Fattorini [144,Chap.5], Henri [200], Miklavcic [284, Chap.5–6], Pazy [330, Chap.8], or Zheng[429]. Let us remark that the mild solution has sometimes alternatively the mean-ing of a limit of the Rothe sequence uτ in C(I;X); cf. Barbu [38, Sect.4.1.1]. Asemigroup approach to Navier-Stokes equations is in Kobayashi and Oharu [234],Miklavcic [284, Sect.6.5] or Sohr [391].

Chapter 10

Evolution governed by certainset-valued mappings

Each of the above presented techniques bears a generalization for the case of set-valued mappings. Now, as in Chapter 5, without narrowing substantially possibleapplications, we will restrict ourselves to the monotonicity method for an initial-value problem for the quite special type of inclusions:

du

dt+ ∂Φ

(u(t)

)+A

(t, u(t)

) � f(t), u(0) = u0, (10.1)

with Φ : V → R ∪ {+∞} a convex potential and A : I × V → V ∗ a Caratheodorymapping such that A is pseudomonotone.

As before, we will first deal with the problem on an abstract level. Thepeculiarity is connected with a possible presence of an indicator function in Φ sothat no growth condition can be assumed on ∂Φ and thus no “dual” estimate onthe time derivative of the solution is at our disposal. So, one must either rely ona regularity or confine oneself to a weak solution which does not involve any timederivative of the solution itself.

10.1 Abstract problems: strong solutions

As a strong solution to (10.1), we will understand u ∈ W 1,∞,2(I;V,H) such that(10.1) holds a.e., in particular u(t) ∈ Dom(Φ) for a.a. t ∈ I.

In view of the definition (5.2) of ∂Φ, i.e. ∂Φ(u(t)) := {ξ ∈ V ∗; ∀v ∈ V :〈ξ, v−u(t)〉+Φ(u(t)) ≤ Φ(v)}, we can write (10.1) in the equivalent form:⟨du

dt+A

(t, u(t)

), v − u(t)

⟩+Φ(v) − Φ

(u(t)

) ≥ ⟨f(t), v − u(t)⟩

(10.2)


335

336 Chapter 10. Evolution governed by certain set-valued mappings

for any v ∈ V . Note that for v �∈ Dom(Φ) this inequality is trivial. A typicalexample is: Φ = ϕ + δK with ϕ : V → R and K ⊂ V convex. Then (10.2) turnsinto the variational inequality for a.a. t ∈ I:

Find u(t)∈K : ∀v∈K :⟨dudt

+A(t, u(t)

), v − u(t)

⟩+ ϕ(v) − ϕ

(u(t)

) ≥ ⟨f(t), v − u(t)⟩. (10.3)

For the special case ϕ = 0, (10.3) can equally be written in the form

du

dt+A

(t, u(t)

) ∈ f(t)−NK

(u(t)

). (10.4)

Thus we have arrived back at (10.1) for a special case ∂Φ = ∂δK = NK .

Lemma 10.1 (Uniqueness). If A satisfies (8.114), i.e. 〈A(t, u)−A(t, v), u− v〉 ≥−c(t)‖u− v‖2H with c ∈ L1(I), then (10.1) has at most one strong solution.

Proof. Take u1, u2 ∈ W 1,∞,2(I;V,H) two strong solutions to (10.1). Put u := u1

and v := u2 into (10.2):⟨du1

dt+A(t, u1), u2−u1

⟩+Φ(u2)− Φ(u1) ≥

⟨f, u2−u1

⟩(10.5)

for a.a. t ∈ I, and analogously for u := u2 and v := u1 we have⟨du2

dt+A(t, u2), u1−u2

⟩+Φ(u1)− Φ(u2) ≥

⟨f, u1−u2

⟩(10.6)

for a.a. t ∈ I. Adding (10.5) and (10.6) and abbreviating u12 = u1 − u2, one gets1

0 ≤ −⟨du12

dt, u12

⟩− ⟨A(t, u1

)−A(t, u2

), u12

⟩ ≤ −1

2

d

dt

∥∥u12

∥∥2H+ c(t)

∥∥u12

∥∥2H.

(10.7)By the Gronwall inequality and by u12(0) = 0, one gets u12 = 0. �

Here, we demonstrate a usage of a regularization method in order to geta sequence of parabolic equations (which we already know how to solve fromChapter 8):

duε

dt+Φ′ε

(uε(t)

)+A

(t, uε(t)

)= f(t) , uε(0) = u0, (10.8)

1As we assume ddt

u∈L2(I;H), we do not have ddt

u∈Lp(I;V ∗) guaranteed if p < 2. However,we certainly have u ∈ L∞(I;H), cf. Lemma 7.1, and thus the first duality in (10.7) can beunderstood as the inner product in L2(I;H) and then Lemma 7.3 can be used for p = 2 andV = H.

10.1. Abstract problems: strong solutions 337

depending on a regularization parameter ε. Let us assume that Φε : V → R isconvex and smooth and satisfies

∀v ∈ W 1,∞,2(I;V,H) : lim supε→0

∫ T

0

Φε

(v(t)

)dt ≤

∫ T

0

Φ(v(t)

)dt, (10.9a)

uε∗⇀ u in W 1,∞,2(I;V,H) ⇒ lim inf

ε→0

∫ T

0

Φε

(uε(t)

)dt ≥

∫ T

0

Φ(u(t)

)dt; (10.9b)

cf. also (5.38).

Theorem 10.2 (A-priori estimates and convergence). Let A be semi-coercivein the sense (8.95) with Z = V and satisfy the growth condition (8.80), andlet (10.9) with Φε ≥ 0 be fulfilled, A(t, ·) : V → V ∗ be pseudomonotone withA(t, v) = A1(v) +A2(t, v) such that

A1 = ϕ′ with some ϕ : V → R, ϕ(v) ≥ c0|v|pV for some c0 ≥ 0, (10.10a)

‖A2(t, v)‖H ≤ γ(t) + C‖v‖p/2V with some γ∈L2(I), C∈R, (10.10b)

u0 ∈ V ∩ Dom(Φ), f ∈ L2(I;H). Then ‖uε‖W 1,∞,2(I;V,H) ≤ C and, for a subse-quence, uε

∗⇀ u in W 1,∞,2(I;V,H) and u is the strong solution to (10.1).

Proof. Let us note that the solution uε to (10.8) does exist.2 The a-priori estimateresults from testing (10.8) by d

dtuε:∥∥∥duε

dt

∥∥∥2H+

d

dtΦε(uε) +

d

dtϕ(uε) =

⟨f −A2(t, uε),

duε

dt

⟩≤ 1

2

∥∥f−A2(t, uε)∥∥2H+

1

2

∥∥∥duε

dt

∥∥∥2H

≤ ∥∥f∥∥2H+ 2γ2+ 2C2

∥∥uε

∥∥pV+

1

2

∥∥∥duε

dt

∥∥∥2H. (10.11)

Then, by using the strategy (8.63)–(8.64) and the Gronwall inequality, one obtainsthe estimate3 ∥∥∥duε

dt

∥∥∥L2(I;H)

≤ C, ‖uε‖L∞(I;V ) ≤ C. (10.12)

Now, take a subsequence uε∗⇀u in W 1,∞,2(I;V,H). As Φε is convex, (10.8) implies⟨duε

dt+A(t, uε(t)), v − uε(t)

⟩+Φε(v) ≥

⟨f(t), v − uε(t)

⟩+Φε

(uε(t)

). (10.13)

Now, we consider v = v(t) with v ∈ Lp(I;V ) ∩ L∞(I;H) and integrate (10.13)over I. Then we can use the usual “parabolic” trick

lim supε→0

⟨duε

dt, v−uε

⟩= lim sup

ε→0

(⟨duε

dt, v⟩− 1

2‖uε(T )‖2H +

1

2‖u0‖2H

)≤⟨dudt

, v⟩− 1

2‖u(T )‖2H +

1

2‖u0‖2H =

⟨dudt

, v − u⟩, (10.14)

2It follows by methods of Chapter 8 by the a-priori estimates derived in (10.11).3Note that Φε(u0) + ϕ(u0) ≤ C < +∞, which follows by (10.9a) from the assumption

u0 ∈ Dom(Φ) ∩ V , and also Φε(uε(T )) + ϕ(uε(T )) ≥ infv∈V,δ>0 Φδ(v) + ϕ(v) ≥ 0.


relying on the fact that uε(T ) ⇀ u(T ) in H because the mapping u �→ u(T ) :W 1,2(I;H) → H is weakly continuous; the by-part integration formula (7.22) isnow backed up by Lemma 7.3 with W 1,2,2(I;H,H) instead of W 1,p,p′

(I;V, V ∗).Furthermore, we can use the test v := u because u ∈ W 1,∞,2(I;V,H), which gives

lim infε→0

∫ T

0

⟨A(t,uε), u− uε

⟩dt ≥ lim inf

ε→0

∫ T

0

⟨f − duε

dt, u− uε

⟩−Φε(u) + Φε(uε) dt ≥ lim inf

ε→0

∫ T

0

⟨f − duε

dt, u− uε

⟩dt

− lim supε→0

∫ T

0

Φε(u) dt+ lim infε→0

∫ T

0

Φε(uε) dt ≥∫ T

0

−Φ(u) + Φ(u) dt = 0;

note that we used both (10.9) and (10.14). Using Lemma 8.29, the obtained pseu-domonotonicity of A yields

lim supε→0

∫ T

0

⟨A(t,uε), v − uε

⟩dt ≤

∫ T

0

⟨A(t,u), v − u

⟩dt. (10.15)

Altogether, using (10.14), (10.15), (10.9a), and (10.9b), we can pass with ε → 0directly in (10.13) integrated over [0, T ], which gives

0 ≤∫ T

0

⟨dudt

− f, v − u⟩+⟨A(t,u), v − u

⟩+Φ

(v(t)

)− Φ(u(t)

)dt. (10.16)

From this, we get 〈 ddtu−f, v−u(t)〉+ 〈A(t,u(t)), v−u(t)〉+Φ(v)− Φ(u(t)) ≥ 0 for

any v ∈ V and for a.a. t ∈ I.4

Moreover, u(0) = u0; realize that certainly uε ⇀ u in C(I;H) and uε(0) = u0.Hence, u is the strong solution to (10.1). �Remark 10.3. In fact, the proof of Theorem 10.2 requires verification of (10.9b)only for a limit u of any subsequence of {uε}ε>0. For example, if K is closed in V ,let us consider5

Φ := δK , Φε(v) :=1

εinfw∈K

‖w − v‖pV . (10.17)

Since (10.11) implies supt∈[0,T ]Φε(uε(t)) ≤ C (as inf ϕ > −∞ is assumed), in the

limit one has u(t) ∈ K for a.a. t. Then∫ T

0Φ(u(t)) dt = 0.6 Since Φε ≥ 0, (10.9b) is

4Assume the contrary, choose a suitable v = v(t) in a measurable (and also integrable) way.5For p = 2, Φ′

ε from (10.17) is the Yosida approximation of ∂Φ = ∂δK = NK(·).6As v �→distV (v,K)p := infw∈K ‖w − v‖pV is certainly continuous on V with a p-growth,

by Theorem 1.43, the corresponding Nemytskiı mapping Lp(I; V ) → L1(I) is continuous and

hence the mapping v �→ ∫ T0 distV (v(t), K)pdt is a continuous functional on Lp(I; V ) which

is convex, hence weakly lower-semicontinuous. As we have uε ⇀ u weakly in Lp(I; V ) and‖distV (uε(·), K)‖L∞(I) = O( p

√ε) thanks to (10.11), we get in the limit∫ T

0distV

(u(t), K

)pdt ≤ lim inf

ε→0

∫ T

0distV

(uε(t), K

)pdt = lim inf

ε→0T O

(p√ε)p

= 0.

10.2. Abstract problems: weak solutions 339

satisfied for this u. The condition (10.9a) is satisfied because, for v(·) �∈ K on some

subset of I with a positive measure, we have trivially lim supε→0

∫ T

0 Φε(v) dt ≤+∞ =

∫ T

0 Φ(v) dt while in the opposite case, i.e. for v(·) ∈ K a.e. on I, we have

limε→0

∫ T

0 Φε(v) dt = limε→0 0 = 0 =∫ T

0 Φ(v) dt.

If K is closed in H , modification of (10.17) by replacing V with H is possible,as well. In this case, the natural option is to consider7

Φ := δK , Φε(v) :=1

εinfw∈K

‖w − v‖2H . (10.18)

Example 10.4. One can apply also the Rothe method, which leads to a sequenceof variational problems at each time level:

ukτ − uk−1

τ

τ+ ∂Φ(uk

τ ) +Akτ (u

kτ ) � fk

τ , u0τ = u0 , (10.19)

with Akτ (u) :=

1τ

∫ kτ

(k−1)τA(t, u) dt and fk

τ :=1τ

∫ kτ

(k−1)τf(t) dt as in (8.81). Existence

of the Rothe approximate solutions can then be shown by Corollary 5.19.

10.2 Abstract problems: weak solutions

As in Definition 8.2, we say now that u ∈ Lp(I;V ) is a weak solution to theinitial-value problem (10.1) if

∫ T

0

⟨dvdt

+A(t, u(t))− f(t), v(t)− u(t)⟩V ∗×V

+ Φ(v(t)

)− Φ(u(t)

)dt ≥ −1

2

∥∥v(0)− u0

∥∥2H

(10.20)

for all v ∈ W 1,p,p′(I;V, V ∗). As in each definition, questions about consistency and

selectivity immediately arise, cf. Sect. 2.4.1. Let us make clear the former one, thelatter being justified by Proposition 10.8 below.

Lemma 10.5. Any strong solution u ∈ W 1,∞,2(I;V,H) to (10.1) is also the weaksolution in the sense (10.20).

Proof. If v = v(t) with v ∈ W 1,p,p′(I;V, V ∗), we have after integration of (10.2)

7In this case, we use ‖distH (uε(t), K)‖L∞(I) = O(√ε). This Φε is Gateaux differentiable.


over I that∫ T

0

⟨dvdt

+A(t, u(t)

)− f(t), v(t) − u(t)⟩+Φ

(v(t)

)− Φ(u(t)

)dt

=

∫ T

0

⟨dudt

+A(t, u(t)

)− f(t), v(t)− u(t)⟩+Φ

(v(t)

)− Φ(u(t)

)+⟨dvdt

− du

dt, v(t) − u(t)

⟩dt ≥ 1

2

∥∥v(T )− u(T )∥∥2H− 1

2

∥∥v(0)− u0

∥∥2H.

(10.21)

This already gives (10.20). �

We also suppose a certain consistency of the operator L = ddt and the “con-

straints” involved implicitly in Φ:

∀u∈Lp(I;V ) ∩ L∞(I;H) ∀u0∈H ∃ a sequence{uδ

}δ>0

⊂ W 1,p,p′(I;V, V ∗) :

lim supδ→0

∫ T

0

Φ(uδ) dt ≤∫ T

0

Φ(u) dt, (10.22a)

u = limδ→0

uδ in Lp(I;V ), (10.22b)

lim supδ→0

∫ T

0

⟨duδ

dt, uδ − u

⟩dt ≤ 0, (10.22c)

u0 = limδ→0

uδ(0) in H. (10.22d)

Theorem 10.6 (A-priori estimates and convergence).8 Let A satisfy thegrowth condition (8.80) and the semicoercivity (8.95) with Z = V and A :Lp(I;V )∩L∞(I;H) → Lp′

(I;V ∗) be pseudomonotone, let Φ′ε satisfy 〈Φ′ε(v), v〉 ≥ 0and the growth condition

∀ε > 0 ∃Cε : R→R increasing ∀v ∈ V :∥∥Φ′ε(v)∥∥V ∗ ≤ Cε(‖v‖H)

(1+‖v‖p−1

V

),

(10.23)moreover f ∈Lp′

(I;V ∗), u0∈H, and let (10.9) be strengthened to

∀v ∈ W 1,p,p′(I;V, V ∗) : lim sup

ε→0

∫ T

0

Φε

(v(t)

)dt ≤

∫ T

0

Φ(v(t)

)dt, (10.24a)

uε∗⇀ u in Lp(I;V ) ∩ L∞(I;H) ⇒ lim inf

ε→0

∫ T

0

Φε

(uε(t)

)dt ≥

∫ T

0

Φ(u(t)

)dt,

(10.24b)

and eventually (10.22) be fulfilled. Then the sequence of weak solutions {uε}ε>0 to(10.8) satisfies ‖uε‖L∞(I;H)∩Lp(I;V ) ≤ C and uε ⇀ u (subsequences) in Lp(I;V )with u being the weak solution to (10.1).

8This assertion is essentially due to Brezis [65], see also Lions [261, Ch.II, Sect.9.3] or Showal-ter [383, Ch.III, Thm.7.1]. For A linear, see also Duvaut and Lions [130, p.51].

10.2. Abstract problems: weak solutions 341

Proof. As shown in Chapter 8, the approximate solution uε∈W 1,p,p′(I;V, V ∗) does

exist. Using a test by uε and by incorporating (8.21), we have the estimate:

1

2

d

dt‖uε‖2H + c0|uε|pV ≤ 1

2

d

dt‖uε‖2H +

⟨Φ′ε(uε)+A(t,uε), uε

⟩+ c1|uε|V + c2‖uε‖2H

= 〈f, uε〉+ c1|uε|V + c2‖uε‖2H≤ C

P

∥∥f∥∥V ∗(|uε|V + ‖uε‖H)+ c1|uε|V + c2‖uε‖2H

≤ ζ(CP+c1)

∣∣uε

∣∣pV+ c1Cζ + C

PCζ

∥∥f∥∥p′

V ∗+(c2+C

P‖f‖V ∗

)(14+‖uε‖2H

)(10.25)

with ζ > 0 which is to be chosen small enough, namely ζ < c0/(CP+c1), and

with CPfrom (8.9) and Cζ depending on p and on ζ like in (1.22). By Gronwall’s

inequality and by (8.9), this gives {uε}ε>0 bounded in L∞(I;H) ∩ Lp(I;V ). Forε > 0 fixed, we can get also the estimate of d

dtuε in Lp′(I;V ∗) because we assumed

a growth condition of the type (8.80) for A + Φ′ε, although not uniformly withrespect to ε > 0, cf. (10.23). Thus we can use the by-part formula (7.22) and, bytesting (10.8) by v − uε, can write

0=

∫ T

0

⟨dvdt

+A(t,uε)−f, v−uε

⟩+ 〈Φ′ε(uε), v−uε〉+

⟨duε

dt−dv

dt, v−uε

⟩dt

≤∫ T

0

⟨dvdt

+A(t,uε)−f, v−uε

⟩+Φε(v)− Φε(uε) dt+

1

2‖v(0)−u0‖2H (10.26)

for any v∈W 1,p,p′(I;V, V ∗). The inequality in (10.26) arose from 〈Φ′ε(uε), v−uε〉 ≤

Φε(v) − Φε(uε) (due to convexity of Φε) and from the obvious inequality 0 ≤12‖v(T )− uε(T )‖2H .

Choosing a subsequence, we have uε∗⇀ u in Lp(I;V ) ∩ L∞(I;H). We are

now to prove (10.15). We cannot put v := u because we do not have the neededinformation d

dtu ∈ Lp′(I;V ∗), hence we must employ the regularization uδ of u

from (10.22). Then, since uδ ∈ W 1,p,p′(I;V, V ∗), we can use (10.26) for v = uδ,

which gives

lim infε→0

∫ T

0

⟨A(t,uε), u−uε

⟩dt = lim inf

ε→0

∫ T

0

⟨A(t,uε), uδ−uε

⟩+⟨A(t,uε), u−uδ

⟩dt

≥ lim infε→0

(∫ T

0

⟨f − duδ

dt, uδ − uε

⟩− Φε(uδ) + Φε(uε) dt

− 1

2

∥∥uδ(0)−u0

∥∥2H− ∥∥A (uε)

∥∥Lp′(I;V ∗)

∥∥u−uδ

∥∥Lp(I;V )

)≥∫ T

0

Φ(u)− Φ(uδ) +⟨duδ

dt− f, u− uδ

⟩dt − 1

2

∥∥uδ(0)− u0

∥∥2H

− lim supε→0

∥∥A (uε)∥∥Lp′(I;V ∗)

∥∥u− uδ

∥∥Lp(I;V )

(10.27)


where (10.24) has been used. Then, passing with δ → 0, by (10.22) and theboundedness of {‖A (uε)‖Lp′(I;V ∗)}ε>0 by (8.80), we can push the right-hand side

of (10.27) to zero, hence we eventually get lim infε→0

∫ T

0〈A(t,uε), u−uε〉dt ≥ 0.

From this, (10.15) follows because A is assumed pseudomonotone from Lp(I;V )∩L∞(I;H) to its (unspecified) dual.

Then, we can estimate from above the limit superior of the right-hand sideof (10.26), which will itself be non-negative, too:

0 ≤ limε→0

∫ T

0

⟨dvdt

− f, v − uε

⟩dt+ lim sup

ε→0

∫ T

0

⟨A(t,uε), v − uε

⟩dt

+ lim supε→0

∫ T

0

Φε(v) dt − lim infε→0

∫ T

0

Φε(uε) dt+1

2

∥∥v(0)− u0

∥∥2H

≤∫ T

0

⟨dvdt

− f +A(t,u), v−u⟩dt+

∫ T

0

Φ(v)−Φ(u) dt+1

2

∥∥v(0)−u0

∥∥2H;

(10.28)

note that we used (10.15) and (10.24). �Remark 10.7. We did not have any information about d

dtu in the preceding Theo-rem 10.6, which is why we could not expect any pseudomonotonicity of A inheritedfrom pseudomonotonicity of A(t, ·) as in Lemmas 8.8 or 8.29. The assumed pseu-domonotonicity of A can be then obtained as, e.g., Example 8.52 or Remark 8.45.

Proposition 10.8 (Uniqueness of the weak solution). Let A(t, ·) be strictlymonotone for a.a. t ∈ I and Φ admit the approximation property (10.22); thenthere is at most one weak solution to (10.1) in the class L∞(I;H).

Proof. 9 Take u1, u2 ∈ Lp(I;V )∩L∞(I;H) two weak solutions, i.e. both u1 and u2

satisfy (10.20). Let us sum (10.20) for u1 and u2, and test it by v ≡ uδ := Rδ(u, u0)with u := 1

2u1+12u2 and with uδ = Rδ(u, u0) denoting a regularization procedure

with the properties (10.22). This gives:

1

2

⟨A (u1)− A (u2), u2 − u1

⟩= lim

δ→0

(⟨A (u1), uδ−u1

⟩+⟨A (u2), uδ−u2

⟩)≥ lim inf

δ→0

(∫ T

0

Φ(u1) + Φ(u2)− 2Φ(uδ) dt

+⟨f − duδ

dt, 2uδ − u1 + u2

⟩− ‖uδ(0)− u0‖2H

)≥ 2 lim inf

δ→0

∫ T

0

Φ(u)− Φ(uδ) dt+ 2 limδ→0

⟨f, uδ − u

⟩− 2 lim sup

δ→0

⟨duδ

dt, uδ−u

⟩− lim

δ→0‖uδ(0)−u0‖2H . (10.29)

9See Lions [261, Chap.II,Sect.9.4] or Showalter [383, Chap.III, Prop.7.1] for p ≥ 2.

10.3. Examples of unilateral parabolic problems 343

Using (10.22) successively to the particular terms we push them to zero for δ → 0.Altogether, this means 〈A (u1)−A (u2), u1 − u2〉 ≤ 0, which gives u1 = u2 by theassumed strict monotonicity of A(t, ·). �Example 10.9 (The regularization procedure (10.22)). Let us illustrate (10.22) fora special case

Φ(u) = ϕ(u) + δK(u) (10.30)

with K ⊂ H convex and closed in H and ϕ : V → R continuous and satis-fying 0 ≤ ϕ(v) ≤ C(1 + ‖v‖pV ). Then, assuming also u0 ∈ K = clH(K ∩V ),we can use the construction (7.19), here with δ in place of ε and with theapproximation u0δ → u0 in H with some {u0δ}δ>0 ⊂ K ∩ V . Obviously, weget uδ ∈ W 1,∞,∞(I;V,H) ⊂ W 1,p,p′

(I;V, V ∗) with the properties (10.22b-d),cf. (7.18a-c). The Nemytskiı mapping Nϕ : Lp(I;V ) → L1(I) is continuous. By

(10.22b),∫ T

0ϕ(uδ(t)) dt →

∫ T

0ϕ(u(t)) dt. Moreover, the convolutory integral (7.19)

remains valued in K if u(t) ∈ K for a.a. t∈I and u0δ ∈ K so that, in particular,uδ from the proof of Lemma 7.4 is valued in K; here we used the convexity of Kand closedness of K in H . Hence, in this case (10.22a) obviously holds because∫ T

0δK(uδ(t)) dt = 0 =

∫ T

0δK(u(t)) dt. If u(t) �∈ K for t from a set in I with a

positive Lebesgue measure, then the right-hand integral in (10.22a) equals +∞and therefore (10.22a) holds in this case, too.

Example 10.10 (The regularization procedure (10.24)). For K ⊂ H closed andΦ = δK , the regularization (10.18) can now be shown to satisfy (10.24) similarlyas we did in Remark 10.3.

10.3 Examples of unilateral parabolic problems

We illustrate the above abstract theory on the evolution variant of the obstacleproblem (5.18) in a special form (i.e. p-Laplacean with zero boundary condition).

Example 10.11 (An obstacle problem: very weak solution). We consider, for w ∈W 1,p(Ω) ∩ L2(Ω) independent of time, the following complementarity problem:

∂u

∂t− div

(|∇u|p−2∇u) ≥ g , u ≥ w,(∂u

∂t− div

(|∇u|p−2∇u)− g

)(u− w) = 0,

⎫⎪⎬⎪⎭ in Q,

∂u

∂ν≥ 0, u ≥ w,

∂u

∂ν(u − w) = 0 on Σ,

u(0, ·) = u0 on Ω.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭(10.31)

The weak formulation results as in (10.20) in a parabolic variational inequality: weseek u(t, ·) ≥ w for a.a. t ∈ I such that∫

Q

(∂v∂t

−g)(

v−u)+∣∣∇u

∣∣p−2∇u ·∇(v−u)dxdt ≥ −1

2

∫Ω

(v(0, ·)−u0

)2dx (10.32)


holds for any v ∈ W 1,p,p′(I;W 1,p(Ω), Lp∗′

(Ω)), v(t, ·) ≥ w for a.a. t ∈ I. Theregularization using a quadratic-penalty method arises as in (10.8) with (10.18):

∂uε

∂t− div

(|∇uε|p−2∇uε

)+

1

ε(uε − w)− = g in Q,

∂u

∂ν= 0 on Σ,

u(0, ·) = u0 on Ω,

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ (10.33)

where v− := min(0, v). Suppose: u0 ∈ L2(Ω), u0 ≥ w, and g ∈ Lp′(I;Lp∗′

(Ω)). Thea-priori estimate can be obtained by multiplication of the equation in (10.33) byuε − w, integration over Ω, and by using Green’s Theorem 1.31:

1

2

d

dt‖uε‖2L2(Ω) + ‖∇uε‖pLp(Ω;Rn) +

1

ε‖(uε − w)−‖2L2(Ω)

=

∫Ω

g(uε − w) +∂uε

∂tw + |∇uε|p−2∇uε∇w dx

≤ N‖g‖Lp∗′(Ω)‖uε−w‖W 1,p(Ω) +

∫Ω

∂uε

∂tw dx+

∥∥|∇uε|p−1∥∥Lp′(Ω)

‖∇w‖Lp(Ω;Rn)

≤ CPN‖g‖Lp∗′(Ω)

(‖∇uε−∇w‖Lp(Ω;Rn) + ‖uε−w‖L2(Ω)

)+

∫Ω

∂uε

∂tw dx+

1

p′‖∇uε‖pLp(Ω;Rn)+

1

p‖∇w‖pLp(Ω;Rn) (10.34)

where N is the norm of the embedding W 1,p(Ω) ⊂ Lp∗(Ω) and C

Pis the constant

from the Poincare inequality (1.55). After absorbing the last-but-one term in theleft-hand side following the strategy (8.168), and making the integration over [0, t],we can use Gronwall’s inequality to estimate:

‖uε(t)‖2L2(Ω) − ‖u0‖2L2(Ω) =

∫ t

0

d

dt‖uε(ϑ)‖2L2(Ω)dϑ ≤ 2

∫ t

0

(‖uε(ϑ, ·)‖2L2(Ω)

+

∫Ω

∂uε

∂tw dx

)dϑ+K

(‖w‖2L2(Ω) + ‖∇w‖pLp(Ω;Rn)

)+K‖g‖p′

Lp′(0,t;Lp∗′(Ω))

with some constant K together with the estimate∫ t

0

∫Ω

∂uε

∂twdxdϑ =

∫Ω

(uε(t)−u0

)wdx ≤ ∥∥uε(t)

∥∥2L2(Ω)

+∥∥u0

∥∥2L2(Ω)

+1

2‖w‖2L2(Ω).

Altogether, this gives the estimates∥∥uε

∥∥L∞(I;L2(Ω))

≤ C, (10.35a)∥∥∇uε

∥∥Lp(Q;Rn)

≤ C, (10.35b)∥∥(uε − w)−∥∥L2(Q)

≤ √εC. (10.35c)


Note that the usual dual estimate ‖ ∂∂tuε‖Lp′(I;W 1,p(Ω)∗) ≤ ε−1/2C → ∞ cannot be

used, hence one cannot expect convergence to a weak solution.However, the convergence to the very weak solution in the sense (10.32) can

then be proved by the methods we used for the weak solution of the abstractproblem in Theorem 10.6 combined with direct treatment of pseudomonotonicityof −Δp by the Minty trick. Let us test the regularized equation (10.33) by v − uε

with v ≥ w, and apply Green’s Theorem 1.31. Realizing that (uε−w)−(v−uε) ≤ 0whenever v ≥ w, it yields∫

Q

(∂uε

∂t− g

)(v − uε) + |∇uε|p−2∇uε · ∇(v − uε) dxdt

= −1

ε

∫Q

(uε − w)−(v − uε) dxdt ≥ 0. (10.36)

Assuming v ∈ W 1,p,p′(I;W 1,p(Ω), Lp′

(Ω)), we can make the by-part integration:∫Q

∂uε

∂t(v − uε) dxdt =

∫Q

∂v

∂t(v − uε) dxdt+

∫Q

(∂uε

∂t− ∂v

∂t

)(v − uε) dxdt

=

∫Q

∂v

∂t(v − uε) dxdt − 1

2‖uε(T, ·)− v(T, ·)‖2L2(Ω) +

1

2‖u0 − v(0, ·)‖2L2(Ω)

≤∫Q

∂v

∂t(v − uε) dxdt +

1

2‖u0 − v(0, ·)‖2L2(Ω)

to obtain∫Q

(∂v∂t

−g)(v−uε) + |∇uε|p−2∇uε·∇(v−uε) dxdt ≥ −1

2‖u0−v(0, ·)‖2L2(Ω).

(10.37)Now we apply the regularization procedure (7.19) which results in uδ(t, x) :=

δ−1∫ +∞0 e−s/δu(t−s, x) ds if u(t, x) is prolonged for t < 0 suitably as in

Lemma 7.4. By the technique (10.27), we obtain lim infε→0

∫Q|∇uε|p−2∇uε ·

∇(u−uε) dxdt ≥ 0. By the monotonicity, boundedness, and radial continuityof the p-Laplacean as a mapping Lp(I;W 1,p(Ω)) → Lp′

(I;W 1,p(Ω)∗), we havealso lim supε→0

∫Q |∇uε|p−2∇uε · ∇(v−uε) dxdt ≤ ∫

Q |∇u|p−2∇u · ∇(v−u) dxdt,

cf. Lemma 2.9. Now we can estimate from above the limit superior in (10.37),which just gives (10.32). Of course, we used also u ≥ 0 implied by (10.35c).

Example 10.12 (An obstacle problem: weak solution). Consider again the problem(10.31) and assume now that g ∈ L2(Q) and u0 ∈ W 1,p(Ω), u0 ≥ w a.e. in Ω, andp > max(1, 2n/(n+2)) so that W 1,p(Ω) � L2(Ω). The weak formulation of theproblem (10.31) requires u(t, ·) ≥ w to satisfy, for any v ≥ w and for a.a. t∈I, theinequality: ∫

Ω

(∂u∂t

− g(t, x))(

v(x)− u(t, x))

+∣∣∇u(t, x)

∣∣p−2∇u(t, x) · (∇v(x) −∇u(t, x))dx ≥ 0 . (10.38)


The needed a-priori estimate (10.12) for the approximate solution uε can be ob-tained by multiplication of the equation in (10.33) by ∂

∂tuε, integration over Ω,and usage of Green’s Theorem 1.31 with the boundary condition in (10.33):

∥∥∥∂uε

∂t

∥∥∥2L2(Ω)

+1

p

∂

∂t

∥∥∇uε

∥∥pLp(Ω;Rn)

+1

2ε

∂

∂t

∥∥(uε − w)−∥∥2L2(Ω)

=

∫Ω

g(t, ·)∂uε

∂tdx ≤ 1

2‖g(t, ·)‖2L2(Ω) +

1

2

∥∥∥∂uε

∂t

∥∥∥2L2(Ω)

. (10.39)

The last term is to be absorbed in the first one. This gives the estimates

∥∥∥∂uε

∂t

∥∥∥L2(Q)

≤ C, (10.40a)∥∥∇uε

∥∥L∞(I;Lp(Ω;Rn))

≤ C, (10.40b)∥∥(uε − w)−∥∥L∞(I;L2(Ω))

≤ √εC. (10.40c)

In view of these estimates, we can suppose that (up to a subsequence) uε → u inL2(Q),10 and also uε ⇀ u in Lp(I;W 1,p(Ω)) and in W 1,2(I;L2(Ω)).

Moreover, the continuity of the Nemytskiı mapping v �→ (v−w)− : L2(Q) →L2(Q), the convergence of uε → u in L2(Q), and (10.40c) yields also∥∥(u − w)−

∥∥L2(Q)

= limε→0

∥∥(uε − w)−∥∥L2(Q)

≤ limε→0

√εC = 0, (10.41)

i.e. u ≥ w for a.a. (t, x) ∈ Q. We want to make a limit passage in (10.36). For p = 2,we can use the concavity of the functional11 u �→ ∫

Q |∇u|p−2∇u · ∇(v−u) dxdt =∫Q ∇u ·∇(v−u) dxdt which, by taking into account its continuity, implies its weak

upper semicontinuity. In general, for p �= 2, we can use the Minty trick (seeLemma 2.13) quite similarly as we did in the steady-state problem, cf. (5.76)–(5.77). For any v ≥ w, by monotonicity of the p-Laplacean and by (10.36), wehave

0 ≥ lim supε→0

∫Q

(|∇uε|p−2∇uε − |∇v|p−2∇v) · ∇(v − uε) dxdt

≥ limε→0

∫Q

(g − ∂uε

∂t

)(v − uε)− |∇v|p−2∇v · ∇(v − uε) dxdt

=

∫Q

(g − ∂u

∂t

)(v − u)− |∇v|p−2∇v · ∇(v − u) dxdt (10.42)

10Here we use Aubin-Lions Lemma 7.7, which gives uε → u in Lγ(I;Lp∗−ε(Ω)) with γ < +∞arbitrary, which is embedded into L2(Q) if p > max(1, 2n/(n+2)).

11Unfortunately, the function a �→ |a|p−2a(b − a) is indeed not concave if p �= 2.


as uε → u in L2(Q).12 Let us put v := (1 − δ)u + δz for z ≥ w a.e.; note thatv ≥ w for any δ ∈ [0, 1]. After dividing it by δ, this gives∫

Q

(g−∂u

∂t

)(z−u)− ∣∣∇u+ δ∇(z−u)

∣∣p−2(∇u+ δ∇(z−u))·∇(z−u) dxdt ≤ 0.

Passing to the limit with δ→0 gives∫Q

(∂

∂tu−g)(z−u) + |∇u|p−2∇u · ∇(z−u) dxdt ≥ 0,

which further gives the point-wise inequality (10.38), cf. Example 8.49.Alternatively, for p ∈ (1,+∞) arbitrary, we can use the fact that the elliptic

part has a potential and transforms the problem into the form∫Q

∂uε

∂t(v − uε) +

1

p|∇v|pdxdt ≥

∫Q

1

p|∇uε|p + g(v − uε) dxdt, (10.43)

and then use the weak lower semicontinuity of u �→ ∫Q |∇u|pdxdt : Lp(Q) → R.

This gives in the limit∫Q

∂u∂t (v−u) + 1

p |∇v|p dxdt ≥ ∫Q

1p |∇u|p + g(v−u) dxdt,

from which already∫Q

∂u∂t (v−u) + |∇u|p−2∇u·∇(v−u) dxdt ≥ ∫

Qg(v−u) dxdt fol-

lows because the convex functional u �→ 1p

∫Ω |∇u|p dx is just the potential of the

mapping A : W 1,p(Ω) → W 1,p(Ω)∗ defined as 〈A(u), v〉 = ∫Ω|∇u|p−2∇u·∇v dx.

Strong convergence in Lp(I;W 1,p(Ω)) can be proved by putting v := u into(10.42), which gives∫

Q

(∣∣∇uε

∣∣p−2∇uε − |∇u|p−2∇u)· ∇(uε − u) dxdt

≤∫Q

(g − ∂uε

∂t

)(uε − u)− |∇u|p−2∇u · ∇(uε − u) dxdt → 0. (10.44)

Then, by the d-monotonicity of the p-Laplacean (cf. Example 2.83) and the uniformconvexity of Lp(I;W 1,p(Ω)), we get strong convergence in this space.

Exercise 10.13. Augment (10.31) by a lower-order term, say c(u), or c(∇u), ordiv(a0(u)) with a0 : R → Rn, and modify Example 10.12 accordingly.

Exercise 10.14. Modify Example 10.12 by considering the unilateral complemen-tarity condition only on Σ as we did in the steady-state case in (5.95).

12Otherwise, we could alternatively use uε(T )⇀u(T ) in L2(Ω) and estimate only “lim inf” by

lim infε→0

∫Q

∂uε

∂tuεdxdt = lim inf

ε→0

1

2‖uε(T )‖2

L2(Ω)− 1

2‖u0‖2L2(Ω)

≥ 1

2‖u(T )‖2

L2(Ω)− 1

2‖u0‖2L2(Ω)

=

∫Q

∂u

∂tudxdt.


Exercise 10.15. Modify (10.42) by using, instead of the monotonicity of the p-Laplacean −Δp, the monotonicity of ∂

∂t −Δp or of −Δp +1ε (· − w)− or of ∂

∂t −Δp +

1ε (· − w)−.

Example 10.16 (Continuous casting: one-phase Stefan problem). In Sect. 5.6.2 wederived the following variational inequality to be satisfied for any v ≥ 0:∫

Ω

3∑i=1

κi∂u

∂xi

∂(v − u)

∂xi+ cv3

∂u

∂x3(v − u) dx+

∫Γ

b(x)u(v − u) dS

≥ − v3

∫Ω

(v − u) dS +

∫Γ

h(v − u) dS (10.45)

for κ1 = κ2 = κ3 = 1. Here we neglect the diffusion flux in the x3-variable (i.e. weput κ3 = 0 while holding κ1 = κ2 = 1) and denote by t = x3/v3 the “residential”time (then v3

∂∂x3

u = ∂∂tu) and L and Ω2 the length and the cross-section of the

casted workpiece. Thus, for T = L/v3, I = [0, T ], Ω = Ω2×I on Figure 14 onp. 167. Now we put Q = I×Ω2 and Σ = I×Γ2 with Γ2 := ∂Ω2 and also use thenotation u(t, x1, x2) (resp. dxdt) instead of u(x1, x2, x3) (resp. dx). Thus (10.45)turns into ∫

Q

∇u · ∇(v − u) + c∂u

∂t(v − u) dtdx+

∫Σ

b(x)u(v − u) dSdt

≥ − v3

∫Q

(v − u) dtdx+

∫Σ

h(v − u) dSdt; (10.46)

of course, now ∇ = ( ∂∂x1

, ∂∂x2

). On the top side we have prescribed the Dirichletboundary condition (cf. again Figure 14) which now turns into the initial condition,while on the bottom side of Γ we now do not prescribe any condition at all. Thuswe arrived at the parabolic variational inequality:

∂u

∂t− div(κ∇u) + v3 ≥ 0, u ≥ 0,(∂u

∂t− div(κ∇u) + v3

)u = 0,

⎫⎪⎬⎪⎭ in Q,

∂u

∂ν+ bu ≥ 0, u ≥ 0,

(∂u∂ν

+ bu)u = 0 on Σ,

u(0, ·) = u0 on Ω2.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭(10.47)

The Baiocchi transformation (5.129) of temperature θ adapted for moving bound-ary problems is called the Duvaut transformation13:

u(t, x1, x2) := −∫ t

0

θ(ϑ, x1, x2)dϑ. (10.48)

13See Duvaut [129]. The new variable is called freezing index; see also Baiocchi [29], Crank[111, Sect.6.4.5], Duvaut-Lions [130, Appendix 3], Rodrigues [354, Sect.2.11].


Exercise 10.17 (Elliptic regularization14). Denote uε the solution to (10.45) withκ1 = κ2 = 1 and κ3 = ε with ε > 0 and show that uε → u in a suitable topologyfor ε ↘ 0, where u solves (10.46). Note that the boundary condition on the bottompart of Γ does not influence the limit.


Evolution variational inequalities and unilateral parabolic problems are addressedby Barbu [37, Sect.IV.3], Elliott, Ockendon [135], Glowinski, Lions, Tremolieres[182], Lions [261, Chap.2,Sect.9 and Chap.3,Sect.6], Naumann [299], Showalter[383], and Zeidler [427, Chap.55]. A fundamental paper is by Brezis [65, Chap.II].Applications to mechanics, in particular to contact problems, is in Duvaut, Lions[130], Eck, Jarusek, Krbec [132], Fremond [154], Hlavacek, Necas [308], Kikuchiand Oden [231].

Variational inequalities in the context of their optimal control are in Barbu[38, Chap.5] or Tiba [404]. For application of Rothe’s method as in Example 10.4we refer to Kacur [219, Section 5.2].

14See Lions [260].

Chapter 11

Doubly-nonlinear problems

In this chapter we touch upon some selected problems not mentioned so far.This concerns situations with the time-derivative d

dtu appearing nonlinearly (Sec-tion 11.1) or acting on a nonlinearity (Section 11.2), in the former case also in

combination with the second time-derivative d2

dt2u involved linearly (Section 11.3).

11.1 Inclusions of the type ∂Ψ( ddtu) + ∂Φ(u) � f

First, we begin with the initial-value problem for the inclusion

∂Ψ(dudt

)+A

(u(t)

) � f(t) , u(0) = u0. (11.1)

As both ∂Ψ and A can be nonlinear and even set-valued (e.g. A = ∂Φ), we speakabout (a special case of) the so-called doubly nonlinear problem. Again, we posethe problem in the framework of Gelfand’s triple V ⊂ H ⊂ V ∗ with compact anddense embeddings.

11.1.1 Potential Ψ valued in R ∪ {+∞}.The first option will simultaneously be an illustration of a technique, not yet

mentioned, based on the test of a differentiated-in-time inclusion by d2

dt2 u. Forthis, we consider A : V → V ∗ in a special form

A = A1 +A2, A1 :V→V ∗ linear, A∗1 = A1, and⟨A1v, v

⟩ ≥ c0|v|2V ,A2 :H→H Lipschitz continuous, (11.2)

and Ψ : H → R ∪ {+∞} uniformly convex on H in the sense

∀ξ1∈∂Ψ(v1), ξ2∈∂Ψ(v2) :⟨ξ1 − ξ2, v1 − v2

⟩ ≥ c1∥∥v1 − v2

∥∥2H

(11.3)


351

352 Chapter 11. Doubly-nonlinear problems

with some c0 and c1 positive, and | · |V again a seminorm on V satisfying theabstract Poincare-type inequality (8.9). The requirement (11.3) implies that onecan write Ψ(v) = Ψ0(v) +

12c1‖v‖2H with some Ψ0 convex, hence also one has

∂Ψ(v) = ∂Ψ0(v) + c1v. (11.4)

On the other hand, we did not impose any growth restriction on Ψ so that, inparticular, Ψ can take values +∞. By (11.2), A1 has a quadratic potential, namelyA1 = Φ′ with Φ(v) = 1

2 〈A1v, v〉, hence as a special case of (11.1) we consider theinclusion

∂Ψ(dudt

)+Φ′

(u(t)

)+A2

(u(t)

) � f(t) , u(0) = u0, (11.5)

with Φ quadratic, and thus smooth, so that ∂Φ(v) = {Φ′(v)}.We will call u ∈ W 1,2(I;V ) a strong solution to (11.1) if u(0) = u0 and the

inclusion in (11.1) is satisfied for a.a. t ∈ I. Equivalently, it means

∀v∈V ∀(a.a.) t∈I :⟨A(u(t)

), v−du

dt

⟩+Ψ(v)−Ψ

(dudt

) ≥ ⟨f(t), v−du

dt

⟩. (11.6)

We will analyze it via the Rothe method, which is now based on the recursiveformula:

∂Ψ(uk

τ − uk−1τ

τ

)+A

(ukτ

) � fkτ , u0

τ = u0, fkτ :=

1

τ

∫ kτ

(k−1)τ

f(t) dt. (11.7)

This determines recursively the Rothe solutions uτ and uτ . As we will need also ananalog of the 2nd-order time derivative, we have to introduce the piecewise affineinterpolation [ d

dtuτ ]i of the piecewise constant time derivative d

dtuτ ; cf. Figure 17on p. 216. This interpolated derivative is defined only on the interval [τ, T ], and itsderivative is obviously piecewise constant and imitates the second-order derivativeof uτ by the following symmetric second-order difference formula:

d

dt

[duτ

dt

]i∣∣∣∣[kτ,(k+1)τ ]

=uk+1τ − 2uk

τ + uk−1τ

τ, k = 1, . . . , T/τ − 1. (11.8)

Proposition 11.1. Let A : V → V ∗ satisfy (11.2) and Ψ : H → R ∪ {+∞} beuniformly convex on H in the sense of (11.3), lower semicontinuous on V , properand ∂Ψ(0) � 0, and V � H. Moreover, let f ∈ W 1,2(I;H) and u0 ∈ V be a steadystate with respect to f(0) in the sense that A(u0) = f(0).

(i) Then the Rothe functions uτ ∈ C(I;V ) and uτ ∈ L∞(I;V ) do exist and wehave the estimates

∥∥uτ

∥∥W 1,∞(I;V )

≤ C,∥∥∥ d

dt

[duτ

dt

]i∥∥∥L2(I;H)

≤ C (11.9)

11.1. Inclusions of the type ∂Ψ( ddtu) + ∂Φ(u) � f 353

for τ sufficiently small, where [·]i denotes the piecewise affine interpolationdefined on the whole interval I by considering formally uk

τ = u0 for k = −1;hence d

dt [ddtuτ ]

i|[0,τ ] := (u1τ − u0)/τ

−2.

(ii) There is a subsequence such that uτ → u weakly* in W 1,∞(I;V ), and everysuch u is a strong solution to (11.1).

To make a-priori estimates, let us first outline the procedure heuristically:assume, for a moment, Ψ ∈ C2(V ) and A2 ∈ C1(H,H), differentiate Φ′( d

dtu) +

A1u + A2(u)= f in time and test it by d2

dt2u, and use symmetry of A1 (so that

〈A1ddtu,

d2

dt2u〉 = 12

ddt 〈A1

ddtu,

ddtu〉):

Ψ′′(dudt

)(d2udt2

,d2u

dt2

)+

1

2

d

dt

⟨A1

du

dt,du

dt

⟩=⟨ d

dt∂Ψ(

du

dt),d2u

dt2

⟩+⟨A1

du

dt,d2u

dt2

⟩=⟨dfdt

−A′2(u)du

dt,d2u

dt2

⟩≤ 1

c1

∥∥∥dfdt

∥∥∥2H+

2

c1

∥∥∥dudt

∥∥∥2H+

c12

∥∥∥d2udt2

∥∥∥2H, (11.10)

where := ‖A′2(·)‖C0(H,L (H,H)) is the Lipschitz constant of A2 : H → H . By(11.3), Ψ′′(·)(ξ, ξ) ≥ c1‖ξ‖2H , hence the first term can be estimated from below

c1‖ d2

dt2u‖2H while the last term is to be absorbed in it. We integrate it over (0, t)

and use 〈A1ddtu,

ddtu〉 ≥ c0| ddtu|2V . We obtain

c02

∣∣∣dudt

(t)∣∣∣2V+

c12

∫ t

0

∥∥∥d2udt2

∥∥∥2Hdt

≤∫ t

0

(1

c1

∥∥∥dfdt

∥∥∥2H+

2

c1

∥∥∥dudt

∥∥∥2H

)dt+

1

2

⟨A1

du

dt(0),

du

dt(0)⟩. (11.11)

Further, we denote U(t) :=∫ t

0 ‖ d2

dϑ2u‖2Hdϑ so that ddtU = ‖ d2

dt2u‖2H and use∥∥∥dudt

(t)∥∥∥2H

=∥∥∥dudt

(0) +

∫ t

0

d2u

dϑ2dϑ∥∥∥2H

≤ 2∥∥∥ ∫ t

0

d2u

dϑ2dϑ∥∥∥2H+ 2

∥∥∥dudt

(0)∥∥∥2H

≤ 2TU(t) + 2∥∥∥dudt

(0)∥∥∥2H

(11.12)

to substitute it into (11.11) to get

c02

∣∣∣dudt

(t)∣∣∣2V+

c12U(t) ≤

∫ t

0

(1

c1

∥∥∥dfdt

∥∥∥2H+

2

c12TU(ϑ)

)dϑ

+ 2T∥∥∥dudt

(0)∥∥∥2H+

1

2

⟨A1

du

dt(0),

du

dt(0)⟩, (11.13)

from which a bound for U(t) uniform in t ∈ I follows by the Gronwall inequal-ity. For t = T it implies a bound for u in W 2,2(I;H) and, putting it again


into (11.11) and using (8.9), also in W 1,∞(I;V ). For using Gronwall’s inequal-ity, 〈A1

ddtu(0),

ddtu(0)〉 must be finite, for which we need the imposed qualification

of u0 with respect to f(0) because ddtu(0) ∈ [∂Ψ]−1(f(0)−A(u0)) = [∂Ψ]−1(0). In

view of the assumption ∂Ψ(0) � 0, we can see that ddtu(0) = 0.

Proof of Proposition 11.1. Seeking u = ukτ satisfying the inclusion (11.7) is equiv-

alent to seeking u solving ∂ϕ(u) +A2(u) � fkτ where

ϕ(v) := τΨ(v − uk−1

τ

τ

)+

1

2

⟨A1v, v

⟩. (11.14)

The existence of such u can be shown by Corollary 5.19; the coercivity followsfrom (11.2) and (11.3) by the estimate1 c0|v|2V +(c1/τ)‖v−uk−1

τ ‖2H ≤ 〈∂ϕ(v), v〉 ≤〈fk

τ −A2(v), v〉 ≤ ‖fkτ ‖H‖v‖H +C(1 + ‖v‖2H) while the pseudomonotonicity of A2

is due to its continuity and the compactness of V � H .

Now, following the strategy (11.10), we are to prove a-priori estimates. Interms of the time-difference

δkτ :=ukτ − uk−1

τ

τ, (11.15)

we modify (11.7) by using (11.4), i.e. c1δkτ + ∂Ψ0(δ

kτ ) + A1u

kτ +A2(u

kτ ) � fk

τ , andwrite it for k and k+1 in the form:⟨

c1δkτ + A1u

kτ + A2(u

kτ ) − fk

τ , v − δkτ⟩

+ Ψ0(v) ≥ Ψ0

(δkτ), (11.16a)⟨

c1δk+1τ +A1u

k+1τ +A2(u

k+1τ )−fk+1

τ , v−δk+1τ

⟩+Ψ0(v) ≥ Ψ0

(δk+1τ

). (11.16b)

As we defined formally u−1τ = u0, we have δ0τ = 0. As A1u0 + A2(u0) = f(0) and

Ψ is minimized at 0 (due to its convexity and the assumption ∂Ψ(0) � 0), theinequality (11.16a) holds for k = 0 as well; one can imagine f extended continu-ously for t < 0 by a constant, hence f0

τ := f(0). We put v := δk+1τ into (11.16a)

and v := δkτ into (11.16b). Adding them for the suggested substitutions and usingthe formula (8.25)2 yields

c1∥∥δk+1

τ − δkτ∥∥2H+

τ

2

⟨A1δ

k+1τ , δk+1

τ

⟩− τ

2

⟨A1δ

kτ , δ

kτ

⟩≤ ⟨c1δk+1

τ − c1δkτ , δ

k+1τ − δkτ

⟩+⟨A1u

k+1τ −A1u

kτ , δ

k+1τ − δkτ

⟩≤ ⟨fk+1

τ − fkτ , δ

k+1τ − δkτ

⟩− ⟨A2(uk+1τ )−A2(u

kτ ), δ

k+1τ − δkτ

⟩≤ 1

c1

∥∥fk+1τ − fk

τ

∥∥2H+ τ2

2

c1

∥∥δk+1τ

∥∥2H+

c12

∥∥δk+1τ − δkτ

∥∥2H. (11.17)

The last term is to be absorbed in the first left-hand-side term. Then, to imitate

1We use ∂Ψ(0) � 0 with (11.3) for ξ2 = 0, v2 = 0, v1 = (v − uk−1τ )/τ .

2It is here τ〈A1δk+1τ , δk+1

τ − δkτ 〉 ≥ τ2〈A1δ

k+1τ , δk+1

τ 〉 − τ2〈A1δkτ , δ

kτ 〉.


(11.11), we sum it for k = 0, . . . , l, l ≤ T/τ , which, after multiplying by 1/τ , gives

1

2

⟨A1δ

l+1τ , δl+1

τ

⟩+ τ

l∑k=0

c12

∥∥∥δk+1τ − δkτ

τ

∥∥∥2H

≤ τl∑

k=0

(1

c1

∥∥∥fk+1τ − fk

τ

τ

∥∥∥2H+

2

c1

∥∥δk+1τ

∥∥2H

)+

1

2

⟨A1δ

0τ , δ

0τ

⟩. (11.18)

Then, after using the discrete analog of (11.12), we use the discrete Gronwallinequality (1.70) provided τ is sufficiently small. The boundedness of the term

τ−1∑l

k=1 ‖fk+1τ − fk

τ ‖2H follows from the assumption f ∈W 1,2(I;H) as in (8.75)-(8.76). Note that 〈A1δ

kτ , δ

kτ 〉 is certainly bounded (as it even vanishes) for k = 0

because δ0τ = 0. Altogether, we get the estimates (11.9). Also,∥∥∥[duτ

dt

]i− duτ

dt

∥∥∥L2(I;H)

=τ√3

∥∥∥ d

dt

[duτ

dt

]i∥∥∥L2(I;H)

= O(τ), (11.19)

cf. (8.50).Then the convergence (in terms of a subsequence) [ ddtuτ ]

i → ddtu in L2(I;H),

which follows by Aubin-Lions Lemma 7.7, implies also ddtuτ → d

dtu in L2(I;H).Also, we have

lim infτ→0

∫ T

0

⟨A1uτ ,

duτ

dt

⟩dt = lim inf

τ→0

1

2

⟨A1uτ (T ), uτ (T )

⟩−1

2

⟨A1u0, u0

⟩+ lim

τ→0

∫ T

0

⟨A1(uτ − uτ ),

duτ

dt

⟩dt

≥ 1

2

⟨A1u(T ), u(T )

⟩− 1

2

⟨A1u0, u0

⟩=

∫ T

0

⟨A1u,

du

dt

⟩dt (11.20)

where we used also3 uτ (T ) ⇀ u(T ) in V and∫ T

0〈A1(uτ − uτ ),

ddtuτ 〉dt = O(τ)

because ‖uτ − uτ‖L2(I;V ) ≤ τ‖ ddtuτ‖L2(I;V ) = O(τ). Then we can make the limit

passage in the equivalent form of (11.7), namely∫ T

0

⟨A1uτ +A2(uτ )− fτ , v − duτ

dt

⟩+Ψ(v)−Ψ

(duτ

dt

)dt ≥ 0 (11.21)

by using

lim infτ→0

∫ T

0

Ψ(duτ

dt

)dt ≥

∫ T

0

Ψ(dudt

)dt (11.22)

because v �→ ∫ T

0 Ψ(v(t)) dt : L∞(I;V ) → R is weakly* lower semicontinuous

and ddtuτ → d

dtu weakly* in L∞(I;V ) due to the estimate in (11.9). Eventually,

3By ddt

uτ ⇀ ddt

u in L2(I; V ) it follows that uτ (T ) = u0+∫ T0

ddt

uτdt ⇀ u0+∫ T0

ddt

udt = u(T ).


A2(uτ ) → A2(u) in L2(I;H), which yields limτ→0

∫ T

0 (A2(uτ ), v − ddtuτ )dt =∫ T

0 (A2(u), v − ddtu) dt. Altogether, (11.21) in the limit results in∫ T

0

⟨A1u+A2(u)− f, v − du

dt

⟩+Ψ(v)−Ψ

(dudt

)dt ≥ 0 (11.23)

from which the pointwise variant (11.6) follows. �

Remark 11.2 (Alternative estimation if A2 = 0). The strategy (11.10)–(11.13) canbe modified if A2 = 0 by estimating∫ t

0

⟨dfdt

,d2u

dt2

⟩dt =

⟨dfdt

(t),du

dt(t)⟩−∫ t

0

⟨d2fdt2

,du

dt

⟩dt−

⟨dfdt

(0),du

dt(0)⟩

≤ CP

∥∥∥dfdt

(t)∥∥∥V ∗

(∣∣∣dudt

(t)∣∣∣V+∥∥∥dudt

(t)∥∥∥H

)+

∫ t

0

CP

∥∥∥d2fdt2

∥∥∥V ∗

(∣∣∣dudt

∣∣∣V+∥∥∥dudt

∥∥∥H

)dt+

∥∥∥dfdt

(0)∥∥∥V ∗

∥∥∥dudt

(0)∥∥∥V

≤ C∥∥∥dfdt

(t)∥∥∥2V ∗

+c04

∣∣∣dudt

(t)∣∣∣2V+

c14U(t)

+

∫ t

0

C∥∥∥d2fdt2

∥∥∥V ∗

(1 +

∣∣∣dudt

∣∣∣2V+ U

)dt+

∥∥∥dfdt

(0)∥∥∥2V ∗

+ C

with C depending on CP, c0, c1, and on ‖du

dt (0)‖V . Usage of Gronwall’s inequalitythen needs f ∈ W 2,1(I;V ∗). Like in Remark 8.23, the Rothe method now needseither usage of the finer version of the discrete Gronwall inequality like in Re-mark 8.15 or a controlled smoothening of f like in (8.15) to be involved in (11.7).Combining it with the previous strategy (11.10)–(11.13), we can thus allow forf ∈ W 1,2(I;H) +W 2,1(I;V ∗).

Example 11.3 (Pseudoparabolic equations). Equations with the time-derivativeinvolved in a (possibly nonlinear) differential operator are sometimes called pseu-doparabolic. An example is the problem with the regularized q-Laplacean:

−div((

ε+ |∇∂u

∂t|q−2

)∇∂u

∂t

)−Δu+ c(u) = g, in Q,

u = 0, on Σ,

u(0, ·) = u0 on Ω,

⎫⎪⎪⎬⎪⎪⎭ (11.24)

with c : R → R Lipschitz continuous and 1 < q ≤ 2. This problem fits withthe above presented theory for V = W 1,2

0 (Ω), H = L2(Ω), Ψ(v) =∫Ω

1q |∇v|q +

ε2 |∇v|2dx, A1 = −Δ, and A2 = Nc; then c1 in (11.3) is εC−2

Pwith C

Pfrom the

Poincare inequality ‖v‖L2(Ω) ≤ CP‖∇v‖L2(Ω;Rn).


Example 11.4 (Parabolic variational inequalities of “type II”4). The above pre-sented abstract theory is fitted to a unilateral constraint acting on the time deriva-tive. An example is the following complementarity problem:

∂u

∂t−Δu+ c(u) ≥ g,

∂u

∂t≥ 0(

∂u

∂t−Δu+ c(u)− g

)∂u

∂t= 0

⎫⎪⎪⎬⎪⎪⎭ in Q,

u = 0 on Σ,

u(0, ·) = u0 on Ω,

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭(11.25)

with c : R → R again Lipschitz continuous. Variational inequality can be obtainedby using the Green formula: find u ∈ W 1,∞(I;L2(Ω)) ∩ L∞(I;W 1,2

0 (Ω)) with∂∂tu ≥ 0 a.e. in Q such that

∀v∈W 1,20 (Ω), v ≥ 0 a.e. on Ω :∫

Ω

(∂u∂t

+ c(u)− g)(

v − ∂u

∂t

)+∇u · ∇

(v − ∂u

∂t

)dx ≥ 0. (11.26)

This fits with the above abstract scheme with Ψ : L2(Ω) → [0,+∞] defined as

Ψ(v) :=

{12‖v‖2L2(Ω) if v ≥ 0 a.e. in Ω

+∞ otherwise,(11.27a)

A1(v) = −Δv, A2(v) = c(v). (11.27b)

Example 11.5 (Boundary inequalities). A unilateral constraint on ∂∂tu can be re-

alized on the boundary Γ. An example is the following complementarity problem:

∂u

∂t−Δu = g, in Q,

∂u

∂ν+ bu ≥ 0,

∂u

∂t≥ 0,

(∂u∂ν

+ bu)∂u∂t

= 0 on Σ,

u(0, ·) = u0 on Ω.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (11.28)

Variational inequality results by the Green formula: find u ∈ W 1,2(I;W 1,2(Ω))with ∂

∂tu|Γ ≥ 0 such that, for all v ∈ W 1,2(Ω) with v|Γ ≥ 0 it holds that∫Ω

(∂u∂t

− g)(

v − ∂u

∂t

)+∇u · ∇

(v − ∂u

∂t

)dx+

∫Γ

bu(v − ∂u

∂t

)dS ≥ 0; (11.29)

we assume u ∈ W 1,2(I;W 1,2(Ω)) to give a good sense of ∇ ∂∂tu and of ∂

∂tu|Γ.

4See Duvant, Lions [130, Chap.II], Glowinski, Lions, Tremoliers [182, Chap.6, Sect. 5] orKacur [219, Sect.5.3] for more information.


This fits with the above abstract scheme with V = W 1,p(Ω), H = L2(Ω), Ψ :W 1,2(Ω) → [0,+∞] defined as

Ψ(v) :=

{ 12‖v‖2L2(Ω) if v|Γ ≥ 0 a.e. on Γ,

+∞ otherwise.(11.30)

If still g ∈ W 1,2(I;L2(Ω)) and u0 qualifies appropriately, by Proposition 11.1,(11.29) has a solution.

11.1.2 Potential Φ valued in R ∪ {+∞}Strengthening the assumptions on Ψ, we can allow the leading part of A, i.e. A1 in(11.2), nonlinear and even set-valued, and also the very restrictive assumption onthe initial condition we made in Proposition 11.1 can thus be put off. Thus, we getanother special case of the doubly nonlinear problem. We confine ourselves to acase when A = A1+A2 with A1 having a convex, possibly nonsmooth (R∪{+∞})-valued potential, let us denote it by Φ, i.e. A1 = ∂Φ, and A2 : H → H . Thus, wehave in mind the inclusion (11.5) with (possibly nonsmooth) Φ : V → R ∪ {+∞}and Ψ : H → R being convex, coercive, and bounded in the sense⟨

∂Ψ(v), v⟩ ≥ c0‖v‖qH − c1‖v‖H , (11.31a)∥∥∂Ψ(v)

∥∥H

≤ C(1 + ‖v‖q−1

H

), (11.31b)⟨

∂Φ(v), v⟩ ≥ c2|v|pV − c3|v|V , (11.31c)∥∥A2(v)

∥∥H

≤ C(1 + ‖v‖q−1

H

); (11.31d)

for some c0, c2 positive, | · |V a seminorm on V satisfying the abstract Poincare-type inequality (8.9), and p, q > 1; in fact, the concrete value of p will not playany role, cf. the estimates (11.35).

By a weak solution to (11.5) we will understand u ∈ W 1,∞,q(I;V,H) suchthat u(0) = u0 ∈ H and, for some w, z ∈ Lq′(I;H), the following system of twoinequalities and one “merging” equality holds:

w + z + A2(u) = f, (11.32a)∫ T

0

⟨w−ξ,

du

dt−v⟩H×H

dt ≥ 0 ∀v∈Lq(I;H), ξ∈Lq′(I;H), ξ∈∂Ψ(v), (11.32b)∫ T

0

⟨z−ξ, u−v

⟩V ∗×V

dt ≥ 0 ∀v∈Lq(I;V ), ξ∈Lq′(I;V ∗), ξ∈∂Φ(v). (11.32c)

The philosophy behind this definition can be seen, without going into details, fromthe fact that (11.32b) means w ∈ ∂Ψ( d

dtu) due to the maximal monotonicity of∂Ψ, cf. Theorem 5.3(ii), and similarly (11.32c) means z ∈ ∂Φ(u). Hence (11.32a)expresses just the inclusion (11.5).


We will analyze it again by Rothe’s method, which consists in the followingrecursive formula:

∂Ψ(uk

τ − uk−1τ

τ

)+ ∂Φ(uk

τ ) +A2(ukτ ) � fk

τ , u0τ = u0, (11.33)

and fkτ from (8.58). This determines recursively the Rothe solutions uτ and uτ ,

and (11.33) for k = 1, . . . , T/τ then means that, for some wτ and zτ piecewiseconstant on the considered partition of I, the following identity and inequalitieshold:

wτ + zτ + A2(uτ ) = fτ , (11.34a)∫ T

0

⟨wτ − ξ,

duτ

dt− v

⟩H×H

dt ≥ 0 ∀v, ξ ∈ L∞(I;H), ξ ∈ ∂Ψ(v), (11.34b)∫ T

0

⟨zτ−ξ, uτ−v

⟩V ∗×V

dt ≥ 0 ∀v∈L∞(I;V ), ∀ξ∈L∞(I;V ∗), ξ∈∂Φ(v).

(11.34c)

In fact, it suffices to require (11.34b,c) to hold for the specified ξ and v piecewiseconstant on the considered partition of I only.

Proposition 11.6 (Colli and Visintin5). Let V � H, the convex, lower semi-continuous functionals Φ : V → R ∪ {+∞} and Ψ : H → R satisfy (11.31a-c),A2 : H → H be continuous and satisfy (11.31d), f ∈ Lq′(I;H) and u0 ∈ dom(Φ);in particular, u0 ∈ V due to (11.31c). Then:

(i) For τ > 0 sufficiently small, uτ ∈C(I;V ) and uτ ∈L∞(I;V ) do exist and thefollowing a-priori estimates hold:∥∥uτ

∥∥W 1,∞,q(I;V,H)

≤ C, (11.35a)∥∥wτ

∥∥Lq′ (I;H)

≤ C, (11.35b)∥∥zτ∥∥Lq′ (I;H)≤ C. (11.35c)

(ii) There is a subsequence and some (u,w, z) ∈ W 1,∞,q(I;V ;H) × Lq′(I;H)2

such that (uτ , wτ , zτ ) converges weakly* to (u,w, z) and any (u,w, z) obtainedin this way satisfies (11.32).

Proof. Existence of the Rothe sequence follows by Corollary 5.19. For this, we de-fine the convex functional ϕ : V → R by v �→ Φ(v)+ τΨ

((v−uk−1

τ )/τ). Then, any

solution to ∂ϕ(u)+A2(u) � fkτ solves also (11.33).6 The needed pseudomonotonic-

ity of A2 : V → V ∗ follows from the compactness of V � H and the continuity of

5See the original work by Colli and Visintin [104] and Colli [101], or also the monograph [418,Sect.III.2] for more details in the special L2-case ∂Φ(u) := −div(a(∇u)).

6Here we use Exercise 5.31.


A2 : H → H , while the coercivity follows from the estimate

12τ

1−qc0‖u‖qH − c1‖u‖H + c2|u|pV − c3|u|V ≤ ⟨∂ϕ(u), u⟩+ Cτ

≤ ⟨fkτ −A2(u), u

⟩+ Cτ ≤ ‖fk

τ −A2(u)‖q′

H + ‖u‖qH + Cτ

≤ 2q′−1‖fk

τ ‖q′

H + 4q′−1Cq′(1 + ‖u‖qH

)+ ‖u‖qH + Cτ , (11.36)

where Cτ is a constant depending on τ and uk−1τ . The last two terms with ‖u‖2H

can be absorbed in the left-hand side if τ is small enough.Let us first outline the a-priori estimate heuristically: test (11.5) by d

dtu andestimate:

c0

∥∥∥dudt

∥∥∥qH− c1

∥∥∥dudt

∥∥∥H+

d

dtΦ(u) ≤

⟨∂Ψ(dudt

),du

dt

⟩+⟨∂Φ(u),

du

dt

⟩≤⟨f −A2(u),

du

dt

⟩≤ 2q

′−1Cε

(‖f‖q′H +

∥∥A2(u)∥∥q′H

)+ ε∥∥∥dudt

∥∥∥qH

≤ 2q′−1Cε

(‖f‖q′H + 2q

′−1Cq′(1 + ‖u‖qH))

+ ε∥∥∥dudt

∥∥∥qH

(11.37)

where c0 and c1 come from (11.31a), C from (11.31d), and where Cε is from (1.22)with q instead of p. The last term is to be absorbed in the first left-hand-side termprovided ε < c0 is chosen. Similarly, the c1-term is to be handled “at the right-hand side” by Young inequality, too. As in (8.64), we denote U(t) :=

∫ t

0‖ ddϑu‖qHdϑ

so that ddtU = ‖ d

dtu‖qH and, by using also

∥∥u(t)∥∥qH

=∥∥∥u0 +

∫ t

0

du

dϑdϑ∥∥∥qH

≤ 2q−1∥∥∥ ∫ t

0

du

dϑdϑ∥∥∥qH+ 2q−1

∥∥u0

∥∥qH

≤ 2q−1tq−1U(t) + 2q−1∥∥u0

∥∥qH

, (11.38)

the estimate (11.37) yields

d

dt

(U +Φ(u)

)≤ C

(‖f(t)‖q′H + U(t))

(11.39)

with some C large enough. Then, by Gronwall’s inequality, we get U(t) + Φ(u(t))bounded independently of t. Then, using also the semi-coercivity7 of Φ and (11.38),we get ‖u(t)‖H and |u(t)|V bounded independently of t, which bounds u inL∞(I;V ) through the Poincare-type inequality (8.9). Eventually, U(T ) < +∞bounds u in W 1,q(I;H).

In the discrete scheme, we test (11.33) by ukτ − uk−1

τ : More precisely, we testwk

τ + zkτ = fkτ − A2(u

kτ ) by δkτ , where wk

τ ∈ ∂Ψ(δkτ ) with δkτ the time difference(11.15) and zkτ ∈ ∂Φ(uk

τ). The last inclusion means Φ(v) ≥ Φ(ukτ ) + 〈zkτ , v−uk

τ 〉.Using v = uk−1

τ and copying the strategy (11.37), we obtain

7The semi-coercivity of Φ means here Φ(v) ≥ c2|v|pV − c3|v|V with some c2 > 0 and followsfrom (11.31c) and the assumed convexity of Φ by Theorem 4.4(i).


c0‖δkτ‖qH +Φ(uk

τ )− Φ(uk−1τ )

τ≤ ⟨wk

τ , δkτ

⟩+⟨zkτ , δ

kτ

⟩+ c1‖δkτ‖H

≤ ⟨fkτ −A2(u

kτ ), δ

kτ

⟩+ c1‖δkτ‖H

≤ 4q′−1Cε

(‖fkτ ‖q

′H + Cq′ + Cq′‖uk

τ‖qH)+ ε∥∥δkτ∥∥qH+ c1

(Cε+ε‖δkτ‖qH

). (11.40)

Taking ε < c0/(1 + c1) and applying the discrete Gronwall inequality (1.70), like(11.38)–(11.39), gives estimate (11.35a).

As wτ ∈ ∂Ψ( ddtuτ ), by (11.31b) we have ‖wτ (t)‖H ≤ C(1 + ‖ d

dtuτ‖q−1H ),

which gives also the estimate (11.35b). Moreover, from zτ = fτ − wτ−A2(uτ ) weget also (11.35c).

As for the limit passage in (11.34), let us choose a subsequence such that:

uτ ⇀ u weakly* in W 1,∞,q(I;V,H), (11.41)

wτ ⇀ w weakly in Lq′(I;H), (11.42)

zτ ⇀ z weakly in Lq′(I;H). (11.43)

From (11.41) and the Aubin-Lions Lemma 7.7, it also follows uτ → u in Lq(I;H).Moreover, we have fτ → f in Lq′(I;H), cf. Remark 8.15. As we have

‖uτ − uτ‖Lq(I;H) =τ

q√q + 1

∥∥∥duτ

dt

∥∥∥Lq(I;H)

= O(τ), (11.44)

cf. (8.50), we have also uτ→u in Lq(I;H), and thus the convergence∫ T

0 〈zτ , uτ 〉dt→ ∫ T

0 〈z, u〉dt in (11.34c) is obvious. Hence z ∈ ∂Φ(u) is proved. Moreover,

A2(uτ ) → A2(u) in Lq′(I;H) due to the continuity of the Nemytskiı mappingA2 : Lq(I;H) → Lq′(I;H), using the continuity of A2 : H → H and the growthcondition (11.31d); cf. Theorem 1.43. The limit passage in (11.34a) is then obvious.As for (11.34b), we use

lim supτ→0

∫ T

0

⟨wτ ,

duτ

dt

⟩dt = lim sup

τ→0

∫ T

0

⟨fτ − A2(uτ )− zτ ,

duτ

dt

⟩dt

≤ limτ→0

∫ T

0

⟨fτ − A2(uτ ),

duτ

dt

⟩dt− lim inf

τ→0

(Φ(uτ (T ))− Φ(u0)

)≤∫ T

0

⟨f − A2(u),

du

dt

⟩dt− Φ

(u(T )

)+Φ(u0)

=

∫ T

0

⟨f − A2(u),

du

dt

⟩− d

dtΦ(u(t)

)dt

=

∫ T

0

⟨f − A2(u)− z,

du

dt

⟩dt =

∫ T

0

⟨w,

du

dt

⟩dt, (11.45)

where the first inequality is based on


∫ T

0

⟨zτ ,

duτ

dt

⟩dt =

T/τ∑k=1

⟨zkτ , u

kτ−uk−1

τ

⟩ ≥ T/τ∑k=1

Φ(ukτ )− Φ(uk−1

τ ) = Φ(uτ (T )

)− Φ(u0),

which follows from convexity of Φ and from that zkτ ∈ ∂Φ(ukτ ), and the second

inequality (11.45) is based on the convergence8 uτ (T ) ⇀ u(T ) and the weak lowersemi-continuity of Φ : V → R∪{+∞}. For the last line in (11.45), we used also thechain rule, namely that d

dtΦ(u) = 〈z, ddtu〉 for any z ∈ ∂Φ(u). This needs rather

subtle arguments:9 For any ε > 0 we can consider a finite partition 0 ≤ tε0 <tε1 < · · · < tεkε

≤ T such that limε→0 maxi=1,...,kε(tεi−tεi−1) = 0 and z is defined at

all tεi and both zε and zRε , defined by zε|(tεi−1,t

εi )

= z(tεi ) and zRε |(tεi−1,t

εi )

= z(tεi−1),

converge to z weakly in Lq′(I;H) for ε → 0. As z(tεi ) ∈ ∂Φ(u(tεi )) for i = 1, . . . , kε,we have⟨

u(tεi )−u(tεi−1), z(tεi−1)

⟩ ≤ Φ(u(tεi )

)−Φ(u(tεi−1)

) ≤ ⟨u(tεi )−u(tεi−1), z(tεi )⟩.

(11.46)

Then, summing it up for i = 1, . . . , kε, we obtain∫ T

0

⟨dudt

, zR

ε

⟩H×H

dt =

kε∑i=1

τεi

⟨u(tεi )−u(tεi−1)

τεi, z(tεi−1)

⟩V×V ∗

≤ Φ(u(tεkε

))− Φ

(u(tε0)

)≤

kε∑i=1

τεi

⟨u(tεi )−u(tεi−1)

τεi, z(tεi )

⟩V×V ∗

=

∫ T

0

⟨dudt

, zε

⟩H×H

dt (11.47)

with τεi := tεi − tεi−1. As before, we have lim infε→0 Φ(u(tεkε)) ≥ Φ(u(T )) because

tεkε→ T and u(tεkε

) ⇀ u(T ) in V . Then, passing ε → 0 in (11.47), we arrive at thelast equality in (11.45).10 �

8This follows from the boundedness of {uτ (T )}τ>0 in V and of { ddt

uτ}τ>0 in Lq(I;H).9In a different way, namely by using Yosida’s regularization of Φ, this chain rule was proved

in [396, Prop. 2.2] even for H a reflexive Banach space similarly as used also in (11.78) below.Note that for a special case Φ = 1

2‖ · ‖2, we stated such a chain rule already in Lemma 9.1 and

then used it in (9.19).10Cf. also Visintin [418, Prop.XI.4.11]. More in detail, both zε and zR

ε are to be understoodas defined equal to zero on I \ [tε0, tεkε

] in (11.79). This is a classical result that we can rely on

limε→0∑kε

i=1(tεi − tεi−1)z(t

εi ) =

∫ T0 z(t) dt for suitable partitions and, as this holds for a.a. parti-

tions, we may also require symmetrically that limε→0∑kε

i=1(tεi − tεi−1)z(t

εi−1) =

∫ T0

z(t) dt. Here

we need rather this result for the Stieltjes-type integral limε→0∑kε

i=1

∫ tεitεi−1

⟨u(t), z(tεi )

⟩dt =∫ T

0

⟨u(t), z(t)

⟩dt for some u ∈ Lq(I;H) fixed, cf. also [116, Lemma 4.12]. Using it for u = d

dtu,

we obtain also the convergence in the two integrals in (11.47). In addition, these partitions can beassumed nested, and we can pick up one common point inside I, and investigate the limit passagein (11.79) separately on the right-hand and the left-hand half-intervals. In the former option, itis like if tε0 would be fixed in (11.79) and then we can see that even limε→0 Φ(u(tεkε

)) = Φ(u(T )).

The analogous argument for the left-hand half-interval then yields limε→0 Φ(u(tε0)) = Φ(u(0)).Let us remark that the technique of replacement of Lebesgue integral by suitable Riemann sumsdates back to Hahn [195] in 1914.


Remark 11.7. Realize that we used only monotonicity of ∂Ψ (not potentiality) forthe basic a-priori estimates. Hencefore, the generalization for a maximal monotonemapping in place of ∂Ψ is possible. Anyhow, the potentiality of ∂Ψ allows foralternative formulation of (11.32b), namely∫ T

0

Ψ(v) +⟨w,

du

dt− v

⟩H×H

dt ≥∫ T

0

Ψ(dudt

)dt (11.48a)

to hold for any v ∈ Lq(I;H). Similarly, (11.32c) bears an alternative formulation∫ T

0

Φ(v) +⟨z, u− v

⟩V ∗×V

dt ≥∫ T

0

Φ(u) dt (11.48b)

to be valid for any v ∈ Lq(I;V ).

Remark 11.8 (Energy balance). The estimate (11.37) has, in concrete motivatedcases, a “physical” interpretation. If Φ is a stored energy and Ψ a (pseudo)potentialof dissipative forces, then 〈∂Ψ( d

dtu),ddtu〉+ d

dtΦ(u) = 〈f, ddtu〉 expresses the balance

between the dissipation rate, the rate of change of stored energy, and the powerof external loading f . Disregarding the non-potential term A2, this balance (as aninequality) is just the core of (11.37).

Proposition 11.9 (Dynamic minimization of Φ). Let f=0 and A2=0, and 11

∃c, α>0 ∀ε>0 ∀v∈H :(ξ∈∂Ψ(v) : ‖ξ‖H ≥ ε

)⇒ ⟨

ξ, v⟩ ≥ cεα. (11.49)

Considering I=[0,+∞) as in Remark 8.22, it holds that limt→+∞ Φ(u(t)) = minΦ.

Proof. 12 Testing (11.5) by ddtu and integrating it over [t1, t2] yields an energy

estimate:

Φ(u(t2)

)− Φ(u(t1)

)+

∫ t2

t1

inf⟨∂Ψ(dudt

(ϑ)),du

dt(ϑ)⟩dϑ ≤ 0; (11.50)

it is to be proved by smoothing of Ψ and a limit passage. Thus13

limt1→+∞t2→+∞

∫ t2

t1

inf⟨∂Ψ(dudt

),du

dt

⟩dt ≤ lim

t1→+∞t2→+∞

(Φ(u(t2)

)− Φ(u(t1)

))= 0, (11.51)

so {∫ t

0inf〈∂Ψ( d

dtu(ϑ)),ddtu(ϑ)〉dϑ}t>0 is Cauchy, hence the limit, denoted by def-

inition∫ +∞0

inf〈∂Ψ( ddtu(ϑ)),

ddtu(ϑ)〉dϑ, does exist and is finite. Put

Iε :={t ∈ [0,+∞); sup

∥∥∥∂Ψ(dudt

)∥∥∥H

< ε}. (11.52)

11The condition (11.49) is satisfied, e.g., for Ψ(v) = ‖v‖qH with q > 1. Then α = q′. In

particular, it holds for the “linear” evolution ddt

u+ ∂Φ(u) � 0 where q = 2. On the other hand,it does not hold for Ψ(v) = ‖v‖H .

12Cf. Aubin and Cellina [28, Chap.3, Sect.4] for the special case Ψ(v) = 12‖v‖2H .

13Here we use that t �→ Φ(u(t)) is bounded from below and, due to (11.50), nondecreasing.


Then the measure of Iε must be infinite, otherwise by (11.49) we would have∫ +∞0

inf〈∂Ψ(dudt (ϑ)),dudt (ϑ)〉dϑ ≥ cεαmeas(R+ \Iε) = +∞, a contradiction. Hence,

for any t∈Iε, we can take ξ∈∂Ψ(dudt ) such that −ξ∈∂Φ(u(t)), and then we have

infϑ>0

Φ(u(ϑ)

) ≤ Φ(u(t)

) ≤ Φ(v) +⟨− ξ, u(t)− v

⟩≤ Φ(v) +

∥∥ξ∥∥H

∥∥u(t)− v∥∥H

≤ Φ(v) + ε∥∥u(t)− v

∥∥H

(11.53)

for v ∈ V ⊂ H arbitrary. Thus infϑ>0 Φ(u(ϑ)) ≤ Φ(v). �

Remark 11.10 (Stefanelli’s variational principle [396]). Considering a fully poten-tial situation (i.e. A2 ≡ 0) and following the idea of the Brezis-Ekeland princi-ple from Sect. 8.10, one can apply two Fenchel inequalities to (11.32) written asw ∈ ∂Ψ( d

dtu) and f−w ∈ ∂Φ(u). Summing it up, this would give the functional

F(u,w) :=∫ T

0Ψ(dudt ) + Ψ∗(w) − 〈w, du

dt 〉 + Φ(u) + Φ∗(f−w) − 〈f−w, u〉dt to be

minimized on W 1,∞,q(I;V,H) × Lq′(I;H) subject to the constraint u(0) = u0.Obviously, F(u,w) ≥ 0. Moreover, F(u,w) = 0 means exactly that w ∈ ∂Ψ( d

dtu)and f−w∈∂Φ(u) hold a.e. on I. Knowing existence of such (u,w) from Proposi-tion 11.6, we can claim existence of such a minimizer. In contrast to (8.247), thepotential F need not be convex. Also its coercivity and, because of the term 〈w, du

dt 〉,weak lower semicontinuity are not obvious. Inspired by (11.45), the latter drawback

can be avoided by substituting∫ T

0 〈w, dudt 〉dt =

∫ T

0 〈f, dudt 〉dt − Φ(u(T )) + Φ(u0).

Yet, it relies on w∈f−∂Φ(u) which is not guaranteed here and which thus mightviolate even the non-negativity of F. Therefore, in [396], the following modificationhas been devised:

F(u,w) :=

(∫ T

0

Ψ(dudt

)+Ψ∗

(w)− ⟨f, du

dt

⟩dt+Φ(u(T ))− Φ(u0)

)+

+

∫ T

0

Φ(u) + Φ∗(f−w) − 〈f−w, u〉dt. (11.54)

Now again F(u,w) ≥ 0 and F(u,w) = 0 means exactly that w ∈ ∂Ψ( ddtu) and

f−w∈∂Φ(u) hold a.e. on I. Like the Brezis-Ekeland principle, F does not involveany derivatives of Φ and Ψ and is especially designed for nonsmooth problems. Itmay serve for the direct method of finding solutions to (11.1) as well as variousmethods of asymptotic analysis, cf. [396].

Exercise 11.11. Modify the proof of Proposition 11.6(ii) for the alternative formu-lation (11.48).

Exercise 11.12 (Decoupling by semi-implicit Rothe method). Consider the systemof two inclusions governed by Φ(u, v), and Ψ(u, v) = Ψ1(u) + Ψ2(v), i.e.

∂Ψ1

(dudt

)+ ∂uΦ(u, v) = f1 and ∂Ψ2

(dvdt

)+ ∂vΦ(u, v) = f2, (11.55a)


and, instead of a fully implicit discretisation, the following semi-implicit formula:

∂Ψ1

(ukτ−uk−1

τ

τ

)+ ∂uΦ(u

kτ , v

k−1τ ) = fk

1,τ , (11.56a)

∂Ψ2

(vkτ−vk−1τ

τ

)+ ∂vΦ(u

kτ , v

kτ ) = fk

2,τ , (11.56b)

and modify the proof of Proposition 11.6, assuming that Φ is only separatelyconvex, i.e. Φ(·, v) and Φ(u, ·) are convex.14

Exercise 11.13 (Evolutionary quasivariational inequality). Consider the u-dependent Ψ = Ψ(u, v) and, instead of (11.1), the inclusion

∂vΨ(u,

du

dt

)+ ∂Φ

(u(t)

) � f(t) , u(0) = u0. (11.57)

Modify the definition (11.48)15 and the proof of Proposition 11.6(ii) for suitablyqualified Ψ.

Exercise 11.14. Like in Exercise 8.62, assuming Φ smooth for simplicity, modifythe estimation scenario (11.37) for A2 having the (possibly nonpolynomial) growth‖A2(u)‖H ≤ C(1+‖u‖q−1

H +Φ(u)1/q′−ε) instead of (11.31d). Assuming ε > 0, mod-

ify also the coercivity estimate (11.36). Finish the proof of Proposition 11.6 byassuming A2 : V1 → H continuous for some V1 � V .

Example 11.15 (Quasistatic friction). Referring to Exercise 11.13, consider V ={v ∈ W 1,2(Ω); v = 0 on ΓD} and the functionals Ψ : V×V → R and Φ : V → Rgiven by

Ψ(u, v) =

∫Ω

1

2|∇v|2 dx+

∫Γ

μb(u+)|v| dS, (11.58a)

Φ(u) =

∫Ω

1

2|∇u|2 dx+

∫ΓN

b(u+) dS (11.58b)

with b(r) = 1α |r|α and b(r) = |r|α−2r with 1 ≤ α < 2#. The test by ∂u

∂t gives thea-priori estimate u ∈ W 1,2(I;W 1,2(Ω)), which also gives compactness of the tracesof ∂u

∂t on ΣN in L2(I;Lα(ΓN)) by combination of the Aubin-Lions theorem withthe trace operator, cf. p. 254. This allows for easy limit passage in the terms as∫ΣN

μb(u+)|∂u∂t | dSdt and∫ΣN

μb(u+)|v| dSdt occurring in the weak formulation.

Such a problem represents a scalar variant of a (regularized) unilateral frictionalcontact problem on ΓN with a friction coefficient μ > 0.

14Hint: To get ukτ , minimize u �→ Φ(u, vk−1

τ ) + τΨ1((u−uk−1τ )τ) − 〈fk

1,τ , u〉. Then minimize

v �→ Φ(ukτ , v) + τΨ2((v−vk−1

τ )τ) − 〈fk2,τ , v〉 get vkτ . For a-priori estimates, use Remark 8.25.

15Hint: instead of (11.48a), consider∫ T0 Ψ(u, v) + 〈w, du

dt− v〉dt ≥ ∫ T

0 Ψ(u, dudt

) dt for any v.


11.1.3 Uniqueness and continuous dependence on data

The doubly-nonlinear structure of (11.1) makes uniqueness of the solution notfully automatic.16 We present two techniques to address this nontrivial task.

Proposition 11.16. Let the assumptions of Proposition 11.1 be fulfilled. Having twosets of data (f, u0) = (fi, u0i) and the corresponding strong solutions ui, i = 1, 2,the following estimate∥∥u1−u2

∥∥W 1,2(I;H)∩L∞(I;V )

≤ C(∥∥f1−f2

∥∥L2(I;H)

+√⟨

A1(u01−u02), u01−u02

⟩ )holds. In particular, it implies uniqueness of the strong solution.

Proof. We write (11.6) modified by using (11.4) for u := u1 and put v := ddtu2,

and also for u := u2 and put v := ddtu1. This gives⟨

c1du1

dt+A1u1(t) +A2(u1),

du2

dt− du1

dt

⟩+Ψ0

(du2

dt

)−Ψ0

(du1

dt

) ≥ ⟨f1(t), du2

dt− du1

dt

⟩, (11.59)⟨

c1du2

dt+A1u2(t) +A2(u2),

du1

dt− du2

dt

⟩+Ψ0

(du1

dt

)−Ψ0

(du2

dt

) ≥ ⟨f2(t), du1

dt− du2

dt

⟩. (11.60)

Summing (11.59) with (11.60), one gets

c1

∥∥∥du1

dt− du2

dt

∥∥∥2H+

1

2

d

dt

⟨A1(u1 − u2), u1 − u2

⟩≤⟨f1 − f2,

du1

dt− du2

dt

⟩−⟨A2(u1)−A2(u2),

du1

dt− du2

dt

⟩≤ 1

c1‖f1 − f2‖2H +

2N2

c1‖u1 − u2‖2V +

c12

∥∥∥du1

dt− du2

dt

∥∥∥2H, (11.61)

from which the claimed estimate follows by Gronwall’s inequality as in (8.63); here stands for the Lipschitz constant of A2 andN for the norm of the embedding V ⊂H . In particular, for f1=f2 and u01=u02, one gets u1=u2, i.e. the uniqueness. �

The nonlinear leading part needs finer technique and additional assumptions.

Proposition 11.17 (Mielke and Theil17). If, in addition to the assumption ofProposition 11.6, A2 = 0 and Φ is uniformly convex and smooth enough so thatΦ′ is strongly monotone and satisfies the Taylor expansion formula∥∥Φ′′(u1)(u2 − u1) + Φ′(u1)− Φ′(u2)

∥∥V ∗ ≤ C

∥∥u1 − u2

∥∥2V, (11.62)

then the solution to (11.5) is unique in the class W 1,1(I;V ).

16For a counterexample see Brokate, Krejcı and Schnabel [70].17See Mielke and Theil [283, Theorem 7.4] for a bit modified case. Later works are by Mielke

[278, Theorem 3.4] and by Brokate, Krejcı and Schnabel [70].

11.2. Inclusions of the type ddtE(u) + ∂Φ(u) � f 367

Proof. Take u1, u2 ∈ W 1,1(I;V ) two solutions to (11.5). We have ∂Ψ( ddtu1) +

Φ′(u1) � f , which means equivalently 〈f − Φ′(u1) − ξ, ddtu1 − v〉 ≥ 0 for any

ξ ∈ ∂Ψ(v). As f − Φ′(u2) ∈ ∂Ψ( ddtu2), we can substitute ξ := f − Φ′(u2) and

v := ddtu2, which gives ⟨

Φ′(u2)− Φ′(u1),du1

dt− du2

dt

⟩≥ 0. (11.63)

Furthermore, put

α(t) :=⟨Φ′(u1)−Φ′(u2), u1−u2

⟩, ri := Φ′′(ui)(ui−u3−i) + Φ′(u3−i)− Φ′(ui)

for i = 1, 2. Then

dα

dt=⟨Φ′′(u1)

du1

dt− Φ′′(u2)

du2

dt, u1 − u2

⟩+⟨Φ′(u1)− Φ′(u2),

du1

dt− du2

dt

⟩=∑i=1,2

⟨Φ′′(ui)(ui − u3−i) + Φ′(ui)− Φ′(u3−i),

dui

dt

⟩=∑i=1,2

⟨ri + 2Φ′(ui)− 2Φ′(u3−i),

dui

dt

⟩.

By (11.62), by strong monotonicity 〈Φ′(u1)− Φ′(u2), u1 − u2〉 ≥ c∥∥u1 − u2

∥∥2V

forsome c > 0, and by (11.63), we can estimate

dα

dt≤ C‖u1 − u2‖2V

(∥∥∥du1

dt

∥∥∥V+∥∥∥du2

dt

∥∥∥V

)+ 2

⟨Φ′(u1)−Φ′(u2),

du1

dt− du2

dt

⟩≤ C

c

(∥∥∥du1

dt

∥∥∥V+∥∥∥du2

dt

∥∥∥V

)α(t) (11.64)

and, from α(0) = 0, we get α ≡ 0 by Gronwall’s inequality. Hence u1 = u2. �Remark 11.18. The assumption (11.62) requires C2,1-smoothness of Φ. Then, theconstant C in (11.62) can be taken as 1

2‖Φ′′‖C0,1(V ;L (V ;V ∗)). For the linear leadingpart, Φ is quadratic and (11.62) is trivially fulfilled with C ≡ 0.

11.2 Inclusions of the type ddtE(u) + ∂Φ(u) � f

Some physically motivated problems lead to double nonlinearity of a structureother than (11.1), namely

dE(u)

dt+A

(u(t)

) � f(t), u(0) = u0. (11.65)

We again consider it posed in the Gelfand triple V � H ∼= H∗ � V ∗. Moreover, wewill consider another Banach space V1 such that V ⊂ V1 ⊂ H (hence H ⊂ V ∗1 ⊂V ∗) and E : V1 → V ∗1 monotone (or possibly even maximal monotone set-valued


E : V1 ⇒ V ∗1 ), and A := A1 + A2 with A1 := ∂Φ, Φ : V → R ∪ {+∞} properconvex, and A2 : V → V ∗. The strong solution is then understood as a couple(u,w) ∈ Lp(I;V )×W 1,∞,p′

(I;V ∗1 , V∗) such that w(0) ∈ E(u0) and

∀v∈Lp(I;V ) :

∫ T

0

Φ(v) +⟨dwdt

+A2(u)−f, v−u⟩V ∗×V

− Φ(u) dt ≥ 0, (11.66a)

∀ξ∈Lq′ (I;V ∗1 ) ∀v∈Lq(I;V1), ξ∈E(v) :

∫ T

0

⟨w−ξ, u−v

⟩V ∗1 ×V1

dt ≥ 0 (11.66b)

with some q > 1. As E : V1 ⇒ V ∗1 is maximal monotone, (11.66b) means w(t) ∈E(u(t)) while, as Φ is convex, (11.66a) means f(t) − d

dtw − A2(u(t)) ∈ ∂Φ(u(t))for a.a. t ∈ I. Hence (11.66) indeed corresponds to (11.65).

11.2.1 The case E := ∂Ψ.

Let us consider the set-valued case E = ∂Ψ with Ψ : V1 → R convex. We apply thenaturally modified Rothe method which now seeks recursively the couple (uk

τ , wkτ ) ∈

V ×V ∗1 such that

wkτ − wk−1

τ

τ+A(uk

τ ) � fkτ , wk

τ ∈ E(ukτ ), (11.67)

for k = 1, . . . , T/τ , and with w0τ ∈ E(u0), u0 ∈ V1.

Lemma 11.19. Assume Ψ : V1→R and A = A1+A2 with A1 = ∂Φ, Φ : V→Rconvex continuous, and A2 : V→V ∗ pseudomonotone. Moreover, if f ∈Lp′

(I;V ∗)and Ψ∗(E(u0)) < +∞,18 and if

∃c0>0, c1, c2∈R :⟨A(v), v

⟩ ≥ c0‖v‖pV −c1‖v‖V −c2‖E(v)‖q′V ∗1, (11.68a)

∃c3∈R : ‖E(v)‖V ∗1≤ c3

(1 + ‖v‖q−1

V1

), (11.68b)

∃C:R → R increasing :∥∥∂Φ(v)∥∥

V ∗ ≤ C(‖E(v)‖V ∗

1

)(1 + ‖v‖p−1

V

), (11.68c)∥∥A2(v)

∥∥V ∗ ≤ C

(‖E(v)‖V ∗1

)(1 + ‖v‖p−1

V

)(11.68d)

for some q > 1, and if τ ≤ τ0 < (q−1)/(q2q−1c2cq−13 ), then there exists the Rothe’s

sequence {uτ}τ>0 and {wτ}τ>0 and∥∥wτ

∥∥L∞(I;V ∗

1 )≤ C,

∥∥uτ

∥∥Lp(I;V )

≤ C,∥∥∥dwτ

dt

∥∥∥Lp′(I;V ∗)

≤ C. (11.69)

Proof. Define B(v) := E(v) + τA(v). Then one can take ukτ = v with v solving

B(v) � τfkτ + wk−1

τ , which does exist by Corollary 5.19,19 and wkτ ∈ E(uk

τ ).

18Legendre-Fenchel’s conjugate Ψ∗ to Ψ is defined Ψ∗(v∗) := supv∈V 〈v∗, v〉−Ψ(v), cf. (8.243).19We use B(v) := ∂Ψ(v) + τ∂Φ(v) + τA2(v) ⊃ ∂[Ψ+ τΦ](v) + τA2(v), cf. Example 5.31. Note

also that, by (11.68b) and (11.73), it holds that 〈w, ∂Ψ∗(w)〉 ≥ ε‖w‖q′V ∗1

for some ε > 0. For

E(u) = w, i.e. u = ∂Ψ−1(w) = ∂Ψ∗(w), we have 〈E(u), u〉 ≥ ε‖E(u)‖q′V ∗1, and therefore, by

(11.68a) and for τ > 0 small enough, the mapping B is coercive as required in Corollary 5.19.


Let us first derive the a-priori estimate heuristically, assuming Ψ∗ and Ψsmooth. Using

d

dtΨ∗(w)=

⟨[Ψ∗]′(w),

dw

dt

⟩V×V ∗

=⟨[Ψ′]−1(w),

dw

dt

⟩V×V ∗

=⟨u,

dw

dt

⟩V×V ∗

(11.70)

with w = Ψ′(u), cf. (8.244), and testing (11.65) by u, we obtain

d

dtΨ∗(w) +

⟨A(u), u

⟩=⟨dwdt

+A(u), u⟩=⟨f, u

⟩. (11.71)

By (11.68a) and by Young’s inequality 〈f, u〉 ≤ Cε‖f‖p′

V ∗ + ε‖u‖pV , it further givesd

dtΨ∗(w) + c0‖u‖pV ≤ c1‖u‖V + Cε‖f‖p

′V ∗ + ε‖u‖pV + c2‖w‖q

′V ∗1

(11.72)

and, taking ε = c0/2, we can make an integration over [0, t] and estimate

(q−1)‖w(t)‖q′V ∗1

q2q−1cq−13

≤ Ψ∗(w(t)

)+

c3q′

≤ Ψ∗(E(u0)

)+ C +

∫ t

0

c2∥∥w(ϑ)∥∥q′

V ∗1dϑ (11.73)

with some C depending on q, c0, c1, c3, and ‖f‖Lp′(I;V ∗), where we used the lower

bound for Ψ∗ obtained from Ψ(v) ≤ c3(1q′ + 2

q ‖v‖qV1) as (8.248)–(8.249).20 By

Gronwall’s inequality, it yields the estimate of w in L∞(I;V ∗1 ). After integration(11.72) over I, we get also the estimate of u in Lp(I;V ). Further, the dual estimateof d

dtw in Lp′(I;V ∗) follows from

sup‖v‖Lp(I;V )≤1

⟨dwdt

, v⟩= sup‖v‖Lp(I;V )≤1

⟨f−A (u), v

⟩ ≤ ∥∥f−A (u)∥∥Lp′(I;V ∗)

≤ ‖f‖Lp′(I;V ∗) + 2C(‖w‖L∞(I;V ∗

1 )

)(∫ T

0

(1+‖u(t)‖p−1

V

)p′dt)1/p′

≤ ‖f‖Lp′(I;V ∗) + 4C(‖w‖L∞(I;V ∗

1 )

)(T 1/p + ‖u‖Lp(I;V )

)p−1. (11.74)

Rigorously, one must proceed by testing (11.67) by ukτ . The difference analog

of (11.70) reads simply as

Ψ∗(wkτ )−Ψ∗(wk−1

τ )

τ≤⟨wk

τ − wk−1τ

τ, uk

τ

⟩, (11.75)

provided ukτ ∈ ∂Ψ∗(wk

τ ), which just follows from the definition of the subdif-ferential, cf. (5.2), or equivalently provided wk

τ ∈ ∂Ψ(ukτ ), cf. (8.244). Then the

difference analog of (11.71)–(11.72) and the discrete Gronwall inequality insteadof (11.73) provided τ ≤ τ0 sufficiently small as specified are simple, as well as theanalog to (11.74). �

20The upper bound for Ψ(v) follows from (11.68b) when one uses the formula (4.6) with

Φ(0) = 0, as we can without loss of generality, which gives Ψ(v) =∫ 10 〈E(tv), v〉dt ≤ ∫ 1

0 c3(1 +‖tv‖V1

)‖v‖V1dt = c3(‖v‖V1

+ ‖v‖qV1/q) ≤ c3(1/q′ + 2‖v‖qV1

/q).


Proposition 11.20. Let, in addition to the assumptions of Lemma 11.19, also V1

be separable, A2 : V → V ∗ be monotone and radially continuous, and V � V1.Then (11.65) possesses a strong solution. Moreover, any weak* limit (u,w) of (asubsequence of) {(uτ , wτ )}τ>0 in Lp(I;V )×W 1,∞,p′

(I;V ∗1 , V∗) solves (11.65).

Proof. In view of (11.67), we have∫ T

0

Φ(v) +⟨dwτ

dt+A2(uτ )− fτ , v − uτ

⟩V ∗×V

− Φ(uτ ) dt ≥ 0, (11.76a)∫ T

0

⟨wτ − ξ, uτ − v

⟩V ∗1 ×V1

dt ≥ 0, (11.76b)

provided ξ(t) ∈ E(v(t)) for a.a. t ∈ I. To obtain (11.66b), we select a subsequenceconverging as indicated, cf. the a-priori estimates (11.69), and pass to the limit in(11.76b) by proving

limτ→0

∫ T

0

⟨wτ , uτ

⟩V ∗1 ×V1

dt = limτ→0

∫ T

0

⟨wτ , uτ

⟩V ∗×V

dt =

∫ T

0

⟨w, u

⟩V ∗1 ×V1

dt. (11.77)

We know that uτ ⇀ u in Lp(I;V ) and, by Aubin-Lions’ compact-embeddinglemma, W 1,∞,p′

(I;V ∗1 , V∗) � Lp′

(I;V ∗), also wτ → w in Lp′(I;V ∗). As d

dtwτ is

bounded in Lp′(I;V ∗), we have ‖wτ − wτ‖Lp′(I;V ∗) = τ‖ d

dtwτ‖Lp′(I;V ∗)/p′√p′+1 =

O(τ), cf. (8.50), and therefore also wτ → w in Lp′(I;V ∗). In this way, (11.77) is

proved. To get (11.66a), we must make a limit passage in (11.76a). We employ

lim infτ→0

∫ T

0

⟨dwτ

dt, uτ

⟩V ∗×V

dt = lim infτ→0

τ

T/τ∑k=1

⟨wkτ − wk−1

τ

τ, uk

τ

⟩V ∗1 ×V1

≥ lim infτ→0

τ

T/τ∑k=1

Ψ∗(wkτ )−Ψ∗(wk−1

τ )

τ

= lim infτ→0

Ψ∗(wτ (T )

)−Ψ∗(w0

)≥ Ψ∗

(w(T )

)− Ψ∗(w0

)=

∫ T

0

⟨dwdt

, u⟩V ∗×V

dt, (11.78)

where (11.75) and (11.70) have been used together with the fact that wτ (T ) ⇀w(T ) in V ∗1 .

21 Moreover, the last equality in (11.78) represents a chain rule for Φ∗

which holds because u(t) ∈ E−1(w(t)) for a.a. t ∈ I has been already proved bylimiting (11.76b) and because Ψ∗ is a potential of E−1. To prove this equality, forany ε > 0 we can consider a finite partition 0 ≤ tε0 < tε1 < · · · < tεkε

≤ T like wedid for (11.45), here guaranteeing that u is defined at all tεi and both uε and uR

ε ,

21Note that {wτ (T )}τ>0 is bounded in V ∗1 due to (11.69) and, by the weak continuity of

W 1,p′(I; V ∗)→ V ∗ : w �→ w(T ), its weak limit in V ∗ (and thus in V ∗1 , too) is just w(T ).


defined by uε|(tεi−1,tεi )

= u(tεi ) and uRε |(tεi−1,t

εi )

= u(tεi−1), converge to u weakly in

Lp(I;V ) for ε → 0. As u(tεi ) ∈ ∂Ψ∗(w(tεi )) for i = 1, . . . , kε, like (11.46), we have⟨w(tεi )−w(tεi−1), u(t

εi−1)

⟩ ≤ Ψ∗(w(tεi )

)−Ψ∗(w(tεi−1)

) ≤ ⟨w(tεi )−w(tεi−1), u(tεi )⟩.

Then, summing it up for i = 1, . . . , kε, we obtain

∫ T

0

⟨dwdt

, uR

ε

⟩V ∗×V

dt =

kε∑i=1

τεi

⟨w(tεi )−w(tεi−1)

τεi, u(tεi−1)

⟩V ∗×V

≤ Ψ∗(w(tεkε

))−Ψ∗

(w(tε0)

)≤

kε∑i=1

τεi

⟨w(tεi )−w(tεi−1)

τεi, u(tεi )

⟩V ∗×V

=

∫ T

0

⟨dwdt

, uε

⟩V ∗×V

dt (11.79)

with τεi := tεi−tεi−1. As before, we have lim infε→0 Ψ∗(w(tεkε

)) ≥ Ψ∗(w(T )) becausetεkε

→ T and w(tεkε) ⇀ w(T ) in V ∗1 . Then, passing ε → 0 in (11.79), we arrive

at the last equality in (11.78) similarly as we did for proving (11.45).22 Using themonotonicity of A2, (11.76a), the convexity of Φ, and (11.78), we obtain

0 ≤ lim supτ→0

∫ T

0

⟨A2(uτ ), uτ − v

⟩− ⟨A2(v), uτ − v⟩dt

≤ lim supτ→0

∫ T

0

Φ(v)− Φ(uτ ) +⟨dwτ

dt−fτ , v−uτ

⟩− ⟨A2(v), uτ−v

⟩dt

≤∫ T

0

Φ(v)− Φ(u) +⟨dwdt

− f +A2(v), v − u⟩dt. (11.80)

Then (11.66a) follows by Minty’s trick, i.e. by putting v := (1−ε)u+ εz, ε ∈ (0, 1],using the convexity of Φ for Φ(v) − Φ(u) ≤ ε(Φ(z)− Φ(u)), dividing it by ε, andpassing to the limit with ε ↘ 0 by using (11.68b,d) and Lebesgue dominated-convergence Theorem 1.14 as in (8.165).

Finally, E(u0) � wτ (0) ⇀ w(0) in V ∗1 and the convexity and closedness ofE(u0) implies w(0) ∈ E(u0). �

Exercise 11.21. Like in (11.48b) relying on potentiality of E = ∂Ψ, one can modify(11.66b) to the variational inequality∫ T

0

Ψ(v) +⟨w, u−v

⟩V ∗1 ×V1

dt ≥∫ T

0

Ψ(u) dt (11.81)

22Note that we cannot directly use that ddt

Ψ∗(w) = 〈u, ddt

w〉V ×V ∗ for any u ∈ ∂Ψ∗(w) ∩ Vlike Lemma 9.1 with Ψ∗ : V1 → R instead of Φ : V → R together with reflexivity of V ∗ (sothat by Komura’s Theorem 1.39 d

dtw is also the strong derivative) because we could assume

Ψ∗ locally Lipschitz continuous on V ∗1 but hardly on V ∗ where w is valued as an absolutely

continuous mapping.


to hold for any v ∈ Lq(I;V1). Modify the proof of Proposition 11.20 for thisalternative definition.23

Exercise 11.22. Considering an abstract Poincare inequality (8.9) modified as‖v‖V ≤ C

P(|v|V + ‖v‖V ∗

1), weaken the semi-coercivity assumption (11.68) as

〈A(v), v〉 ≥ c0|v|pV −c1|v|V −c2‖E(v)‖q′V ∗1

and modify the estimate (11.72) accord-

ingly.

Remark 11.23. A general drawback of the technique based on the test by u isthat it does not yield any information about du

dt , which prevented us from usingPapageorgiou’s Lemma 8.8 and a possibly non-monotone A2. Note that, formally,dudt = d

dtE−1(w) = [E−1]′(w)dwdt but [E−1]′(w) ∈ L (V ∗1 , V1) while dw

dt is valuedonly in V ∗ but not in V ∗1 . In some specific cases, [E−1]′(w) admits an extensionon V ∗, which may open a way to estimate du

dt ; compare E = identity on H usedin Chapters 8 and 10.

11.2.2 The case E non-potential

An alternative approach to analysis of (11.65) is based on testing by ddtu. It allows

us to consider Φ unbounded and valued in R∪{+∞} and to replace the assumptionon potentiality of E by its uniform monotonicity. In addition, assuming E smoothwith E′ : V1 → V ∗1 bounded, one can use the semi-implicit Rothe method:

E′(uk−1τ )

ukτ − uk−1

τ

τ+A(uk

τ ) � fkτ (11.82)

for k = 1, . . . , T/τ , u0τ = u0. This, linearizing partly the problem, can lead to ad-

vantageous numerical strategies after applying additionally the Galerkin method.

Lemma 11.24. Let A = A1+A2 with A1 = ∂Φ with Φ : V → R∪{+∞} convexlower semicontinuous, and let f ∈ L2(I;V ∗1 ), u0 ∈ dom(Φ), and

∃c1 > 0 ∀u, v∈V1 :⟨E′(u)v, v

⟩V ∗1 ×V1

≥ c1‖v‖2V1, (11.83a)

∃c0 > 0 ∀v∈V : Φ(v) ≥ c0|v|pV , (11.83b)

∃c2∈R ∀v∈V :∥∥A2(v)

∥∥V ∗1≤ c2

(1 + ‖v‖V1

)(11.83c)

with some p > 0 (whose value is not reflected in (11.84)) and with | · |V re-ferring to a seminorm satisfying a modified Poincare inequality (8.9), namely‖v‖V ≤ C

P(|v|V +‖v‖V1). Then, for τ > 0 sufficiently small, the following a-priori

estimates hold: ∥∥uτ

∥∥L∞(I;V )

≤ C,∥∥∥duτ

dt

∥∥∥L2(I;V1)

≤ C. (11.84)

23Hint: Instead of (11.76b), use∫ T0Ψ(v) + 〈wτ , uτ−v〉V ∗

1 ×V1dt ≥ ∫ T

0Ψ(uτ ) dt, and make the

limit passage by weak lower semicontinuity of u �→ ∫ T0 Ψ(u) dt and again by (11.77).


Proof. Let us first proceed heuristically: testing (11.65) by ddtu, using that

〈 ddtE(u), d

dtu〉 = 〈E′(u) ddtu,

ddtu〉 ≥ c1‖ d

dtu‖2V1, and integrating it over [0, t] gives

Φ(u(t))− Φ(u0) + c1U(t) ≤∫ t

0

d

dϑΦ(u) +

⟨E′(u(ϑ))

du

dϑ,du

dϑ

⟩V ∗1 ×V1

dϑ

=

∫ t

0

⟨dE(u)

dϑ+ Φ′(u(ϑ)),

du

dϑ

⟩dϑ =

∫ t

0

⟨f(ϑ)−A2(u(ϑ)),

du

dϑ

⟩dϑ

≤∫ t

0

c12

∥∥∥dudϑ

∥∥∥2V1

+1

c1‖f(ϑ)‖2V ∗

1+ 2

c22c1

(1 + ‖u(ϑ)‖2V1

)dϑ

≤ c12U(t) +

1

c1‖f‖2L2([0,t];V ∗

1 )+ 2Tc22c1

(1 + 2‖u0‖2V1

+ 2

∫ t

0

U(ϑ) dϑ), (11.85)

where the last estimate follows as (8.63) and, as before, U(t) :=∫ t

0‖ ddϑu‖2V1

dϑ. ByGronwall’s inequality (1.66) and by (11.83b) together with the abstract Poincare-type inequality (8.9), it yields the estimate of u in L∞(I;V ) and of d

dtu in L2(I;V1).Rigorously, the a-priori estimate (11.84) can be obtained by testing (11.82)

by (ukτ − uk−1

τ )/τ :

c1


τ

τ

∥∥∥2V1

+Φ(uk

τ )−Φ(uk−1τ )

τ≤ Φ(uk

τ )−Φ(uk−1τ )

τ

+⟨E′(uk−1

τ )(uk

τ−uk−1τ

τ

),ukτ−uk−1

τ

τ

⟩≤⟨fkτ −A2(u

kτ ),

ukτ−uk−1

τ

τ

⟩(11.86)

and by continuing the strategy (11.85) by the discrete Gronwall inequality. �Proposition 11.25. Let, in addition to the assumptions of Lemma 11.24, also E′ :V1 → L (V1, V

∗1 ) be continuous and bounded in the sense ‖E′(v)‖L (V1,V ∗

1 ) ≤ C(1+‖v‖qV1

) for some q > 1, A2 : V1 → V ∗1 be continuous and V � V1. Then (11.65)possesses a strong solution. Moreover, any (u,w), with w = E(u) and u a weak*limit of (a subsequence of) {uτ}τ>0 in W 1,∞,2(I;V, V1), solves (11.65).

Proof. Choosing a convergent subsequence uτ∗⇀ u in W 1,∞,2(I;V, V1), we make

a limit passage in

E′(uR

τ )duτ

dt+ ∂Φ(uτ ) +A2(uτ ) � fτ , (11.87)

with the “retarded” Rothe function uRτ as defined in (8.202). Using V � V1 and

Aubin-Lions’ lemma, we have uτ → u in Lq1(I;V1) for any q1 < +∞. Then, as‖uτ − uR

τ ‖L2(I;V1) = τ‖ ddtuτ‖L2(I;V1)/

√3 = O(τ), cf. (8.50), we have uR

τ → u inL2(I;V1) and, by the interpolation between L2(I;V1) and L∞(I;V1), also uR

τ → uin Lq1(I;V1). By the same arguments, also uτ → u in Lq1(I;V1). By continuity ofthe Nemytskiı mapping induced by E′ as Lq1(I;V1) → Lq1/q(I;L (V1, V

∗1 )), and

by ddtuτ ⇀ d

dtu in L2(I;V1), we can see that E′(uRτ )

ddtuτ converges to E′(u) d

dtu

weakly in L2q1/(2q+q1)(I;V ∗1 ). Then 〈E′(uRτ )

ddtuτ , uτ 〉 → 〈E′(u) d

dtu, u〉 providedwe choose q1 ≥ 2q + 2. By (11.83c), we have the Nemytskiı mapping induced by


A2 continuous as Lq1(I;V1) → Lq1(I;V ∗1 ), hence A2(uτ ) → A2(u) in Lq1(I;V ∗1 ).Then, using also the convexity of Φ, we can pass to the limit in (11.87) written inthe form (11.66a), i.e. we can make limit superior in∫ T

0

Φ(v) +⟨E′(uR

τ )duτ

dt+A2(uτ )− fτ , v − uτ

⟩V ∗1 ×V1

− Φ(uτ ) dt ≥ 0. (11.88)

This gives just (11.66a) when realizing E′(u) ddtu = d

dtE(u) and putting w = E(u).Then (11.66b) follows, too. Eventually, the limit passage in the initial conditionu0 = uτ (0) ⇀ u(0) yields u(0) = u0. �

Remark 11.26. Like in Remark 8.23, a more general f ∈ L2(I;V ∗1 ) +W 1,1(I;V ∗)can be considered. Also, like in Exercise 8.62, a modification of the growth condi-tion (11.83c) as ‖A2(v)‖V ∗

1≤ c2(1 + ‖v‖V1 +Φ(v)1/2) can also be considered.

Example 11.27 (Heat equation). The nonlinear heat equation in the form (8.199)can be transformed by considering β(u) as the unknown function itself intothe form ∂

∂t

(β−1(u)

) − Δu = g with the boundary condition ∂∂νu + (b1 +

b2| κ−1β−1(u)|3) κ−1(β−1(u)) = h. Assuming β : R → R increasing and satis-fying |β−1(r)| ≤ C(1 + |r|q−1) for some q ≥ 2, we can apply the approach fromSections 11.2.1-2 with V := W 1,2(Ω), V1 := Lq(Ω), H := L2(Ω). Even (11.68b)allows for β : R → R to be only non-decreasing, which causes the problem to beelliptic on regions where the values of the solution u range over the interval(s)where β is constant. Thus one can treat it as a so-called parabolic/elliptic equa-tion, here arising from the phenomenon that the heat capacity c in (8.197) decaysto zero for some values of temperature. Then one speaks about a degenerate heatequation.

Example 11.28 (Landau-Lifschitz-Gilbert equation [179, 251]). In fact, this sec-tion 11.2.2 applies to the inclusions of the type B(u)dudt + A(u) � f even in the

cases when B(u)dudt cannot be reduced to ddtE(u) for any E. An example of such

a structure is a pseudoparabolic equation

α∂u

∂t− β(|u|)u × ∂u

∂t− μΔu+ c(u) = g, (11.89)

where u is R3-valued, × : R3×R3 → R3 is the vector product of two 3-dimensionalvectors, α, μ > 0, β : R → R is continuous, and c : R3 → R3 is continuous with acoercive potential. Such an equation describes evolution of a magnetization vectoru in magnetic materials.24 Considering further an initial-boundary-value problem

24The minima of the potential of c determines directions of easy magnetisation, the so-calledgyromagnetic β-term causes non-dissipative precession of the magnetisation vector, while theα-term determines attenuation of this precession, the μ-term is the so-called exchange energywhich is of quantum origin, and g is the external magnetisation.


for (11.89), the semi-implicit discretisation of the type (11.82) yields

αukτ−uk−1

τ

τ− β(|uk−1

τ |)uk−1τ ×uk

τ−uk−1τ

τ− μΔuk

τ + c(ukτ ) = gkτ . (11.90)

Exercise 11.29. Prove existence of a weak solution ukτ ∈ W 1,2(Ω;R3) to (11.90)

with a suitable boundary condition (e.g. ∂∂νu

kτ = 0 on Γ), then prove a-priori

bounds for uτ in L∞(I;W 1,2(Ω;R3))∩W 1,2(I;L2(Ω;R3)) provided g ∈ L2(Q;R3)and the initial and boundary data are qualified appropriately, and eventually proveconvergence for τ → 0 to a weak solution to (11.89).25

11.2.3 Uniqueness

The notion of monotonicity of A can be generalized to be suitable for the doubly-nonlinear structure (11.65) with E : V1 → V ∗1 invertible such that E−1(E(V )) ⊂V : the mapping A : V → V ∗ is called E-monotone (in Gajewski’s sense [164]) if

∀u, v∈V, z = E−1(E(u)+E(v)

2

):

⟨A(u), u−z

⟩+⟨A(v), v−z

⟩ ≥ 0. (11.91)

If, in addition, strict inequality in (11.91) implies u = v, then A is called strictlyE-monotone. Obviously, if E is linear, then (strict) E-monotonicity is just theconventional (strict) monotonicity.

In case that E = Ψ′ and A is E-monotone, the function

�(u, v) := Ψ∗(E(u)

)+Ψ∗

(E(v)

) − 2Ψ∗(E(u)+E(v)

2

), (11.92)

introduced by Gajewski [164], can measure a “distance” of two solutions u1 andu2 corresponding to two initial conditions u01 and u02 in the sense that, for allt ∈ I, the mapping u0 �→ u(t) is non-expansive:

�(u1(t), u2(t)

) ≤ �(u01, u02

). (11.93)

Indeed, as in the last equality in (11.78) with t instead of T , we have

Ψ∗(E(u(t))

)−Ψ∗(E(u(0))

)=

∫ t

0

⟨u(ϑ),

dE(u(ϑ))

dϑ

⟩dϑ. (11.94)

Then, using subsequently the definition (11.92) and the formula (11.94) with z(t) =E−1(12E(u1(t)) +

12E(u2(t))) and assuming a special case f ≡ 0, we obtain

�(u1(t), u2(t)

)− �(u01, u02

)=

∫ t

0

⟨u1(ϑ),

dE(u1(ϑ))

dϑ

⟩25Hint: For existence of uk ∈ W 1,2(Ω;R3), use coercivity on the particular time levels; realize

the cancellation (uk−1τ ×uk)·uk = 0. For a-priori estimates, execute the test by uk

τ−uk−1τ and

realize the cancellation (uk−1τ ×(uk

τ−uk−1τ ))·(uk

τ−uk−1τ ) = 0. For convergence use uτ → u in

L2(Q;R3) by the Rellich-Kondrachov Theorem 1.21 and also uRτ − uτ → 0 in L2(Q;R3).


+⟨u2(ϑ),

dE(u2(ϑ))

dϑ

⟩− 2

⟨ d

dϑ

E(u1(ϑ)) + E(u2(ϑ))

2, z(ϑ)

⟩dϑ

=

∫ t

0

⟨u1(ϑ)− z(ϑ),

dE(u1(ϑ))

dϑ

⟩+⟨u2(ϑ)− z(ϑ),

dE(u2(ϑ))

dϑ

⟩dϑ

= −∫ t

0

⟨A(u1(ϑ)), u1(ϑ)−z(ϑ)

⟩+⟨A(u2(ϑ)), u2(ϑ)−z(ϑ)

⟩dϑ ≤ 0. (11.95)

Proposition 11.30 (Gajewski, Groger, Necas26). Let E = Ψ′ be invertible, Ψbe convex, and let:

(i) A be strictly E-monotone and f = 0, or

(ii) A = ∂Φ with both Φ and Φ◦E−1 convex, E be strongly monotone in the sense

〈E(u)−E(v), u−v〉V ∗1 ×V1 ≥ m

∥∥u−v∥∥2V1

with m > 0 and Lipschitz continuous,

[E−1]′ : V ∗1 → L (V ∗1 , V1) be Lipschitz continuous, and f ∈L1(I;V ∗1 ).Then the inclusion (11.65) admits at most one strong solution.

Proof. In case (i), (11.95) implies �(u1(t), u2(t)

)=0, thus u1(t)=u2(t) for a.a. t∈I.

As to (ii), abbreviating w = E(u), (11.65) can be written as the system oftwo inclusions:

dw

dt+ ∂Φ(u) � f, (11.96a)

du

dt+ [E−1]′(w)∂Φ

(E−1(w)

) � [E−1]′(w)f. (11.96b)

Note that ddtw = d

dt(E(u)) = E′(u) ddtu. Let us consider two solutions (u1, w1)

and (u2, w2), write (11.96) for them, subtract the particular inclusions in (11.96)tested by u1 − u2 and w1 −w2, respectively, and sum it up. Altogether, using alsothe assumed monotonicity of A and of ∂[Φ ◦ E−1],27 we get

d

dt

⟨w1 − w2, u1 − u2

⟩V ∗1 ×V1

=⟨dw1

dt− dw2

dt, u1 − u2

⟩V ∗×V

+⟨du1

dt− du2

dt, w1−w2

⟩V ∗1 ×V1

≤ ⟨([E−1]′(w1)− [E−1]′(w2))f, w1−w2

⟩V ∗1 ×V1

≤ L∥∥w1 − w2

∥∥2V ∗1

∥∥f∥∥V ∗1≤ LM2

∥∥u1 − u2

∥∥2V1

∥∥f∥∥V ∗1

where L and M are the Lipschitz constants of [E−1]′ and of E, respectively.Integrating over I, one obtainsm‖u1(t)−u2(t)‖2V1

≤ 〈w1(t)−w2(t), u1(t)−u2(t)〉 ≤LM2

∫ t

0

∥∥u1(ϑ)− u2(ϑ)∥∥2V1

∥∥f(ϑ)∥∥V ∗1dϑ, from which u1(t) = u2(t) a.e. on I follows

by Gronwall’s inequality. �26The case (i) is basically due to Gajewski [164] while the case (ii) has earlier been investigated

by Groger and Necas [192] in a narrower setting V = V1 = H.27The convexity of Φ ◦ E−1 implies E-monotonicity of ∂Φ.

11.3. 2nd-order equations 377

11.3 2nd-order equations

We will now treat the abstract 2nd-order doubly-nonlinear Cauchy problem (9.63)in a non-autonomous variant, i.e.:

d2u

dt2+A

(t,du

dt

)+B

(t, u(t)

)= f(t) , u(0) = u0 ,

du

dt(0) = v0. (11.97)

Enough dissipation (i.e. A uniformly monotone on V ) causes that (11.97) repre-sents, in fact, a parabolic problem in terms of “velocity” d

dtu.

By the strong solution we will understand u ∈ W 2,∞,p,p′(I;V, V, V ∗), cf. the

notation (7.4), such that (11.97) holds for a.a. t ∈ I.Considering finite-dimensional subspaces Vk of Z ⊂ V satisfying (2.7) with

Z from (8.95), we apply the Galerkin method with u0k ∈ Vk approximating u0

in V and v0k ∈ Vk approximating v0 in H . Existence of a solution uk of theresulting initial-value problem for the system of ordinary-differential equationscan be proved by the usual prolongation technique combined with the followinga-priori estimates.

Lemma 11.31 (A-priori estimates). Let A : I × V → V ∗ be semi-coercive inthe sense of (8.95), let B : V → V ∗ be time-independent and have a potentialΦ : V → R+, and let f ∈ Lp′

(I;V ∗), u0k, v0k ∈ Vk, limk→∞ u0k = u0 in V andlimk→∞ v0k = v0 in H. Then, with C independent of k,∥∥∥duk

dt

∥∥∥L∞(I;H)∩Lp(I;V )

≤ C & ‖uk‖L∞(I;V ) ≤ C. (11.98)

Moreover, if A satisfies the growth conditions (8.80), i.e.

‖A(t, u)‖V ∗ ≤ C(‖u‖H)(γA(t) + ‖u‖p−1V ),

and B satisfies ‖B(u)‖V ∗ ≤ C(‖u‖V ) with γA ∈ Lp′(I) and C increasing, then∣∣∣d2uk

dt2

∣∣∣p′,l

≤ C (11.99)

for any k ≥ l; for the seminorm | · |p′,l see (8.94).

Proof. For (11.98), let us test the equation d2

dt2uk + A ( ddtuk) + B(uk) = fk by

ddtuk, and use 〈 d2

dt2uk,ddtuk〉 = 1

2ddt‖ d

dtuk‖2H , cf. also (7.23). Using also a strategylike that used in (8.21), we get

1

2

d

dt

∥∥∥duk

dt

∥∥∥2H+ c0

∣∣∣duk

dt

∣∣∣pV+

d

dtΦ(uk) ≤ c1(t)

∣∣∣duk

dt

∣∣∣V+ c2(t)

∥∥∥duk

dt

∥∥∥2H

+⟨f,

duk

dt

⟩≤ Cεc

p′1 + ε(t)

∣∣∣duk

dt

∣∣∣pV+ c2(t)

∥∥∥duk

dt

∥∥∥2H

+ CεCP

∥∥f∥∥p′

V ∗ + εCP

∣∣∣duk

dt

∣∣∣pV+ C

P

∥∥f∥∥V ∗

(12+

1

2

∥∥∥duk

dt

∥∥∥2H

). (11.100)


For ε > 0 small enough, we get (11.98) by the Gronwall inequality. As {u0k}k∈N ⊂V is bounded and { d

dtuk}k∈N ⊂ Lp(I;V ) is bounded, by Lemma 7.1 with V1 =V2 = V we get even {uk}k∈N ∈ C(I;V ) bounded. The dual estimate (11.99) canbe obtained from∣∣∣d2uk

dt2

∣∣∣p′,l

:= sup‖z‖Lp(I;V )≤1

z(t)∈Vl for a.a. t∈I

⟨d2uk

dt2, z⟩=⟨f − A (

duk

dt)− B(uk), z

⟩

≤ sup‖z‖Lp(I;V )≤1

∥∥∥f + A(duk

dt

)+ B(uk)

∥∥∥Lp′(I;V ∗)

‖z‖pLp(I;V )

≤ ‖f‖Lp′(I;V ∗)+ C(∥∥duk

dt

∥∥L∞(I;H)

)×(‖γA‖Lp′(I) +

∥∥∥duk

dt

∥∥∥p−1

Lp(I;V )

)+ T 1/p′

C(‖uk‖L∞(I;V )

). (11.101)

�

Lemma 11.32 (Other estimates). Let

A = A1 +A2 with A1 time-independent,

A1 = Ψ′ for a convex potential Ψ : V → R,

Ψ(v) ≥ c0‖v‖pV , c0 > 0,

‖A2(t, v)‖H ≤ C1(t) + C2‖v‖p/2V , C1 ∈ L2(I), (11.102)

B = B1 +B2 with B1 : V → V ∗ smooth (and time-independent),∥∥B1(u)∥∥V ∗ ≤ C3

(1 + ‖u‖p−1

V

),⟨

[B′1(u)](v), v⟩ ≤ C4

(1 + ‖u‖p−2

V

)‖v‖2V ,∥∥B2(t, u)∥∥H

≤ C5(t) + C6‖u‖p/2V , C5 ∈ L2(I), (11.103)

with p ≥ 2 if B1 �= 0, and let f ∈ L2(I;H), u0k, v0k ∈ Vk and now bothlimk→∞ u0k = u0 and limk→∞ v0k = v0 in V . Then∥∥∥duk

dt

∥∥∥L∞(I;V )

≤ C &∥∥∥d2uk

dt2

∥∥∥L2(I;H)

≤ C. (11.104)

Proof. For (11.104), we test the Galerkin equation by d2

dt2uk. Using (11.102), onegets ∥∥∥d2uk

dt2

∥∥∥2H+

d

dtΨ(duk

dt

)=⟨f(t)−A2

(t,duk

dt

)−B(t, uk),

d2uk

dt2

⟩≤ 2

∥∥f(t)∥∥2H+ 2

∥∥∥A2

(t,duk

dt

)∥∥∥2H−⟨B1(uk),

d2uk

dt2

⟩+2∥∥B2

(t, uk

)∥∥2H+

1

2

∥∥∥d2uk

dt2

∥∥∥2H


≤ 2∥∥f(t)∥∥2

H+ 4C2

1 (t) + 4C22

∥∥∥duk

dt

∥∥∥pV−⟨B1(uk),

d2uk

dt2

⟩+4C2

5(t) + 4C26

∥∥uk

∥∥pV+

1

2

∥∥∥d2uk

dt2

∥∥∥2H. (11.105)

Absorbing the last term in the left-hand side and integrating this estimate over[0, t] and using the coercivity of Ψ assumed in (11.102) and the by-part formula∫ t

0〈B1(uk),

d2

dϑ2 uk〉dϑ = 〈B1(uk(t)),ddtuk(t)〉− 〈B1(u0k), v0k〉−

∫ t

0〈[B′1(uk)](

ddϑuk),

ddϑuk〉dϑ, we get

c0

∥∥∥duk

dt

∥∥∥pV+

∫ t

0

∥∥∥d2uk

dϑ2

∥∥∥2Hdϑ ≤ Ψ

(duk

dt(t))+

∫ t

0

∥∥∥d2uk

dϑ2

∥∥∥2Hdϑ

≤ Ψ(v0k) +

∫ t

0

(2∥∥f(ϑ)∥∥2

H+ 4C2

1 (ϑ) + 4C22

∥∥∥duk

dϑ

∥∥∥pV

+⟨[

B′1(uk)](duk

dϑ

),duk

dϑ

⟩+ 4C2

5 (ϑ) + 4C26

∥∥uk(ϑ)∥∥pV

)dϑ

−⟨B1

(uk(t)

),duk

dt(t)⟩+⟨B1(u0k), v0k

⟩≤ Ψ(v0k) + 2‖f‖2L2(I;H) + 4‖C1‖2L2(I) + 4‖C5‖2L2(I)

+

∫ t

0

(4C2

2

∥∥∥duk

dϑ

∥∥∥pV+C4+2C4

∥∥∥duk

dϑ

∥∥∥pV+(C4+4C2

6 )∥∥uk(ϑ)

∥∥pV

)dϑ

+ CεCp′3

(1 + ‖uk(t)‖p−1

V

)p′+ ε∥∥∥duk

dt(t)∥∥∥pV+⟨B1(u0k), v0k

⟩; (11.106)

note that p≥2 was needed to apply Holder’s inequality if B1 �=0. For ε<c0 we canabsorb the last-but-one term in the left-hand side. As in (11.38), we can estimate

∥∥uk(t)∥∥pV≤ (2t)p−1Uk(t)+2p−1‖u0k‖pV , with Uk(t) :=

∫ t

0

∥∥∥duk

dϑ

∥∥∥pVdϑ. (11.107)

Hence the estimate (11.106) exhibits the structure∥∥∥duk

dt

∥∥∥pV+

∫ t

0

∥∥∥d2uk

dϑ2

∥∥∥2Hdϑ ≤ C+C

∫ t

0

(∥∥∥duk

dϑ

∥∥∥pV+Uk(ϑ)

)dϑ+CUk(t) (11.108)

with a sufficiently large constant C. Adding (C+1)Uk(t) = (C+1)∫ t

0‖ ddϑuk‖pV dϑ,

we obtain∥∥∥duk

dt

∥∥∥pV+

∫ t

0

∥∥∥d2uk

dϑ2

∥∥∥2Hdϑ+ Uk(t) ≤ C +

∫ t

0

((2C+1)

∥∥∥duk

dϑ

∥∥∥pV+ CUk(ϑ)

)dϑ,

which eventually allows us to use Gronwall’s inequality to conclude the bound for‖ ddtuk(t)‖V uniformly for t ∈ I as well as the second estimate in (11.104). �


Theorem 11.33 (Convergence). Suppose V � H and the assumptions ofLemma 11.31 hold (so that the a-priori estimates (11.98)–(11.99) are at our dis-posal), and one of the following situations holds:

(i) A = A1 +A2, with A1 is linear, time-independent, symmetric (i.e. A∗1 = A1),and positive semi-definite, A2 continuous as a mapping Lq(I;H) → Lp′

(I;V ∗)with some q < +∞, and B : V → V ∗ is semi-coercive and pseudomonotone.

(ii) A(t, ·) : V → V ∗ is pseudomonotone for a.a. t ∈ I and B = B1 + B2

with B1 : V → V ∗ linear, monotone and symmetric in the sense B∗1 = B1,B2 : W 1,p(I;V ) → Lp′

(I;V ∗) totally continuous, and, for simplicity, u0 ∈ V1

(hence u0 ∈ Vk for any k > 1, too).

Then uk converges (as a subsequence) to a strong solution to (11.97).

Proof. Take z∈W 1,p,p′(I;V, V ∗) and a sequence {zk}k∈N, zk ∈W 1,∞(I;Vk), such

that zk → z in W 1,p,p′(I;V, V ∗lcs); by a density argument it does exist.28 We use

zk as a test function for (11.97). By the by-part integration, we obtain∫ T

0

−⟨duk

dt,dzkdt

⟩+⟨A(duk

dt

), zk

⟩+⟨B(uk), zk

⟩− ⟨f, zk⟩dt+⟨duk

dt(T ), zk(T )

⟩=⟨v0k, zk(0)

⟩. (11.109)

Let us now choose a subsequence such that

uτ ⇀ u in W 1,p(I;V ) and uτ∗⇀ u in W 1,∞(I;H). (11.110)

Then also

uk(T ) = u0 +

∫ T

0

duk

dtdt ⇀ u0 +

∫ T

0

du

dtdt = u(T ) in V , (11.111)

hence uk(T ) → u(T ) in H � V . By (11.99) and by the interpolated Aubin-LionsLemma 7.829, we then have { d

dtuk}k∈N relatively compact in Lq(I;H) with anyq < +∞. Therefore,

duk

dt→ du

dtin Lq(I;H). (11.112)

In particular, ddtuk → d

dtu in L2(I;H). Moreover, by the L∞(I;H)-estimate of

{ ddtuk}k∈N, choosing (for a moment only) a subsequence, d

dtuk(T ) converges weakly

in H . By (11.99), ddtuk(T ) =

∫ T

0d2

dt2ukdt+ v0k → ∫ T

0d2

dt2udt+ v0 in V ∗lcs, hence

duτ

dt(T ) ⇀

du

dt(T ) in H. (11.113)

28Cf. the proof of Lemma 8.28 with the arguments in the proof of Theorem 8.31.29We use Lemma 7.8 here with V1 = V , V2 = V4 = H, and V3 = V ∗

lcs.


Putting zk − uk instead of zk into (11.109), one gets

lim infk→∞

⟨B(uk), zk − uk

⟩= lim inf

k→∞

(∫ T

0

⟨f, zk − uk

⟩+⟨duk

dt,dzkdt

− duk

dt

⟩dt

−∫ T

0

⟨A1

duk

dt, zk − uk

⟩dt−

⟨A2

(duk

dt

), zk − uk

⟩−⟨duk

dt(T ), zk(T )− uk(T )

⟩+ 〈v0k, zk(0)− u0k〉

)=

∫ T

0

⟨f, z−u

⟩+⟨dudt

,dz

dt− du

dt

⟩dt− lim sup

k→∞

∫ T

0

⟨A1

duk

dt, zk−uk

⟩dt

−⟨A2

(dudt

), z − u

⟩−⟨dudt

(T ), z(T )− u(T )⟩+ 〈v0, z(0)− u0〉

because ddtuk → d

dtu in L2(I;H), ddtzk → d

dtz in L2(I;H), and we also used

(11.111) together with V � H and (11.113). The term with A2 uses ddtuk → d

dtu

in Lq(I;H) and the assumption that A2 : Lq(I;H) → Lp′(I;V ∗) is continuous.

By (11.111) and by the weak upper semi-continuity of z �→ −〈A1z, z〉 : V →R, one gets

lim supk→∞

⟨A 1

duk

dt, zk − uk

⟩= lim

k→∞

∫ T

0

⟨A1

duk

dt, zk

⟩dt− lim inf

k→∞

⟨A1uk(T ), uk(T )

⟩2

+

⟨A1u0, u0

⟩2

≤∫ T

0

⟨A1

du

dt, z⟩dt−

⟨A1u(T ), u(T )

⟩2

+

⟨A1u0, u0

⟩2

=⟨A 1

du

dt, z−u

⟩.

(11.114)

Then

lim infk→∞

⟨B(uk), zk−uk

⟩ ≥ ∫ T

0

⟨f, z−u

⟩+⟨dudt

,dz

dt− du

dt

⟩dt

−⟨A (

du

dt), z−u

⟩−⟨dudt

(T ), z(T )⟩+⟨v0, z(0)

⟩. (11.115)

We have {B(uk)}k∈N bounded in Lp′(I;V ∗) (cf. the assumptions in Lemma 11.31)

and zk → z in Lp(I;V ), so that

lim infk→∞

⟨B(uk), z−uk

⟩= lim

k→∞⟨B(uk), z−zk

⟩+ lim inf

k→∞⟨B(uk), zk−uk

⟩= lim inf

k→∞⟨B(uk), zk−uk

⟩. (11.116)

In particular, for z := u we have lim supk→∞〈B(uk), uk − u〉 ≤ 0 and, in viewof Lemma 8.29, we can use the pseudomonotonicity of B to conclude that, for


any z ∈ Lp(I;V ), lim infk→∞〈B(uk), uk − z〉 ≥ 〈B(u), u − z〉. Joining it with

(11.115) and (11.116), one gets 〈B(u), u−z〉 ≤ ∫ T

0 〈f, u−z〉−〈 ddtu,

ddtz− d

dtu〉dt−〈A ( d

dtu), u− z〉− 〈 ddtu(T ), z(T )−u(T )〉+〈v0, z(0)−u(0)〉. As it holds for any z, we

can conclude that⟨B(u), z

⟩=

∫ T

0

⟨f−A

(t,du

dt

), z⟩−⟨dudt

,dz

dt

⟩dt−

⟨dudt

(T ), z(T )⟩+⟨v0, z(0)

⟩.

(11.117)Moreover, the initial conditions d

dtu(0) = v0 and u(0) = u0 are satisfied by

the continuity arguments. As z ∈ W 1,p,p′(I;V, V ∗) we can use the formula (7.15)

for z(T ) = 0 = z(0), which enables us to rewrite (11.117) into the form (11.97).In the case (ii), we use the pseudomonotonicity of the mapping A +B ◦L−1

where [L−1v](t) =∫ t

0v(ϑ) dϑ + u0 is the inverse mapping to L = d

dt : dom(L) →Lp(I;V ) with

dom(L) ={u∈W 1,p(I;V ); u(0) = u0

}, (11.118)

cf. (8.227). Note that uk is the Galerkin approximation to (11.97) if and only ifvk = L−1uk and 〈 d

dtvk +[A +B ◦L−1](vk)− f, zk〉 = 0 for any zk ∈ Lp(I;Vk) andvk(0) = v0k; here u0 ∈ Vk has been employed. By Lemma 8.29, A is pseudomono-tone on W 1,p,p′

(I;V, V ∗lcs) ∩ L∞(I;H). The mapping v �→ B1(L−1v) : Lp(I;V ) →

C(I;V ) ⊂ Lp′(I;V ∗) is monotone: indeed, for v1, v2 ∈ Lp(I;V ) we again abbre-

viate v12 = v1 − v2 and then, using the symmetry and monotonicity of B1, wehave⟨

B1(L−1v1)−B1(L

−1v2), v1−v2⟩=

∫ T

0

⟨B1

(∫ t

0

v12(ϑ)dϑ), v12(t)

⟩dt

=

∫ T

0

⟨B1

( ∫ t

0

v12(ϑ) dϑ),d

dt

∫ t

0

v12(ϑ) dϑ⟩dt

=1

2

∫ T

0

d

dt

⟨B1

(∫ t

0

v12(ϑ) dϑ),

∫ t

0

v12(ϑ) dϑ⟩dt

=1

2

⟨B1

(∫ T

0

v12(ϑ) dϑ),

∫ T

0

v12(ϑ) dϑ⟩− 1

2

⟨B10, 0

⟩ ≥ 0. (11.119)

Moreover, B1 ◦L−1 : Lp(I;V ) → Lp′(I;V ∗) is bounded as both L−1 : Lp(I;V ) →

L∞(I;V ) and B1 : V → V ∗ are bounded. Hence, it is also radially continuous. ByLemma 2.9, B1 ◦ L−1 is pseudomonotone. Also, L−1 : Lp(I;V ) → W 1,p(I;V ) isweakly continuous, and B2 : W 1,p(I;V ) → Lp′

(I;V ) is assumed totally continu-ous, B2 ◦ L−1 : Lp(I;V ∗) → Lp′

(I;V ∗) is totally continuous. Altogether, due toLemma 2.11(i) and Corollary 2.12, A +B1 ◦L−1 +B2 ◦L−1 is pseudomonotone.Then we can employ Theorem 8.30 with A +B◦L−1 in place of A to get v solvingddtv + A (v) + B(L−1v) = f and v(0) = v0. Then it suffices to put u = L−1v. �

Remark 11.34 (Energy balance). Testing the equation (11.97) by ddtu, which leads

to the a-priori estimate (11.100), has in concrete motivated cases a “physical”


interpretation. If Φ is a potential of B (cf. Lemma 11.31) then, integrating over[0, t], this test leads to

1

2

∥∥∥dudt

(t)∥∥∥2H+Φ

(u(t)

)︸︷︷︸

total energy at time t

+

∫ t

0

⟨A(dudϑ

),du

dϑ

⟩dϑ︸︷︷︸

dissipated energy

=1

2

∥∥v0∥∥2H +Φ(u0

)︸︷︷︸total energy at time 0

+

∫ t

0

⟨f(ϑ),

du

dϑ

⟩dϑ︸︷︷︸

work of external forces

(11.120)

which just expresses the balance of “mechanical” energy. Here, the total energymeans the sum of the “kinetic” energy ‖ d

dtu‖2H and the “stored” energy Φ(u),cf. also (12.11) below.

Proposition 11.35 (Uniqueness30). Let A be “weakly monotone” in the sense of(8.114) and one of the following situations takes place:

(i) B is linear of the form B(t, u) = B1u with B1 : V → V ∗ monotone andsymmetric, i.e. B∗1 = B1.

(ii) B is Lipschitz continuous on H, i.e. ‖B(t, u)−B(t, v)‖H ≤ (t)‖u−v‖H with ∈ L2(I).

(iii) ‖B(t, u)−B(t, v)‖p′/2V ∗ ≤ (t)‖u−v‖H with ∈ L2(I) and 〈A(t, u)−A(t, v), u−

v〉 ≥ c0‖u − v‖pV − c1(t)‖u − v‖V − c2(t)‖u − v‖2H with c0 > 0, c1 ∈ Lp′(I),

and c2 ∈ L1(I).

Then (11.97) possesses at most one (strong) solution.

Proof. We take two solutions u1 and u2, subtract (11.97) for u = u1 and u = u2,and test it by v = d

dtu12 where u12 := u1 − u2. We thus get

1

2

d

dt

(∥∥∥du12

dt

∥∥∥2H+⟨B1(u12), u12

⟩)=

1

2

d

dt

∥∥∥du12

dt

∥∥∥2H+⟨B(t, u1)−B(t, u2),

du12

dt

⟩= −

⟨A(t,du1

dt

)−A(t,du2

dt

),du12

dt

⟩≤ c(t)

∥∥∥du12

dt

∥∥∥2H.

By the Gronwall inequality and by u12(0) = 0 and ddtu12(0) = 0, one gets u1 = u2.

In the case (ii), we can estimate

1

2

d

dt

∥∥∥du12

dt

∥∥∥2H

= −⟨A(t,

d

dtu1)−A(t,

d

dtu2),

du12

dt

⟩+⟨B(t, u2)−B(t, u1),

du12

dt

⟩≤ c(t)

∥∥∥du12

dt

∥∥∥2H+∥∥B(t, u1)−B(t, u2)

∥∥2H+∥∥∥du12

dt

∥∥∥2H

(11.121)

≤ c(t)∥∥∥du12

dt

∥∥∥2H+ (t)2

∥∥u12

∥∥2H+∥∥∥du12

dt

∥∥∥2H

30For the case (iii), cf. also Zeidler [427, Chap.33].


where c(·) comes from (8.114). Abbreviating still ddtu12 = v12, we get

1

2

d

dt‖v12‖2H ≤ c(t)‖v12‖2H + (t)2‖u12‖2H + ‖v12‖2H . (11.122)

Multiplying ddtu12 = v12 by u12, one gets

1

2

d

dt‖u12‖2H =

⟨v12, u12

⟩ ≤ 1

2‖v12‖2H +

1

2‖u12‖2H . (11.123)

Now we apply the Gronwall inequality to the system (11.122)–(11.123) togetherwith u12(0)=0 and v12(0)=0, which gives, in particular, that u12(t)=0 for all t.

In the case (iii), we get analogously

1

2

d

dt‖v12‖2H + c0‖v12‖pV ≤ c1(t)‖v12‖V + c2(t)‖v12‖2H+⟨B(t, u2)−B(t, u1), v12

⟩ ≤ Cεc1(t)p′+ c2(t)‖v12‖2H

+ Cε‖B(t, u1)− B(t, u2)‖p′

V ∗ + ε‖v12‖pV . (11.124)

We choose ε < c0 to absorb the last term and also the estimate ‖B(t, u1) −B(t, u2)‖p

′V ∗ ≤ (t)2‖u12‖2H . Then we apply again the Gronwall inequality to the

system (11.123)–(11.124). �

Remark 11.36. Velocity ∂∂tu is indeed a natural test function, as already claimed

in Remark 11.34, while u itself is not a suitable test function here. This is re-lated to troubles typically arising in variational inequalities with obstacles likeu ≥ 0, i.e. B = ∂Φ with Φ = δK , K := {v ≥ 0}. E.g., after a penaliza-

tion, one can consider ∂2

∂t2uε − Δ ∂∂tuε +

1εu−ε = g. By testing by ∂

∂tuε, one gets

‖ ∂∂tuε‖L∞(I;L2(Ω))∩L2(I;W 1,2(Ω)) = O(1) and ‖u−ε ‖L∞(I;L2(Ω)) = O(

√ε). But, we

must test by v − uε to prove convergence, and we get after the by-part integra-tion the term

∫Q | ∂∂tuε|2 with a “bad” sign. Note that we do not have the “dual

estimate” to ∂2

∂t2uε uniform in ε. Also, a test by ∂2

∂t2 uε yields the penalty term

ε−1u−ε∂2

∂t2uε which cannot be estimated “on the left-hand side”.

Remark 11.37 (Rothe method). The semidiscretization in time is also here applica-ble to (11.97): we define uk

τ ∈ V , k = 1, . . . ,K, by the following recursive formula:

ukτ − 2uk−1

τ + uk−2τ

τ2+Ak

τ

(ukτ − uk−1

τ

τ

)+Bk

τ (ukτ ) = fk

τ , (11.125a)

u0τ = u0, u−1

τ = u0 − τv0, (11.125b)

where again fkτ :=

1τ

∫ kτ

(k−1)τf(t) dt and Ak

τ (u) :=1τ

∫ kτ

(k−1)τA(t, u) dt and Bk

τ (u) :=

1τ

∫ kτ

(k−1)τB(t, u) dt. Existence of the Rothe sequence then needs, as in Lemma 8.5,

11.4. Exercises 385

Akτ and Bk

τ to be pseudomonotone and semi-coercive for k = 1, . . . , T/τ . Variousmodifications of the above procedure are as usual. Instead of (11.99) or (11.104),we need here to estimate d

dt [ddtuτ ]

i in Lp′(I;V ∗) or in L2(I;H), where [·]i denotes

the piecewise-linear interpolation operator, cf. Figure 17 on p.216, defined now onthe whole interval I = [0, T ] because, thanks to (11.125b), we defined the Rothesequence {uk

τ} even for k = −1. By using the discrete by-part summation formula

T/τ∑k=1

⟨uk−2uk−1+uk−2, zk

⟩=⟨uT/τ−uT/τ−1, zT/τ

⟩− ⟨u0−u−1, z1⟩

−T/τ∑k=2

⟨uk−1−uk−2, zk−zk−1

⟩(11.126)

and considering a test function z ∈ W 1,p,p′(I;V, V ∗), one obtains an analog of

(11.109), namely∫ T

0

⟨A(duτ

dt

), zτ

⟩+⟨B(uτ ), zτ

⟩− ⟨fτ , zτ⟩dt−∫ T

τ

⟨duτ

dt(· − τ),

dzτdt

⟩dt+

⟨duτ

dt(T ), zτ (T )

⟩=⟨v0, zτ (τ)

⟩(11.127)

where zτ and zτ are respectively the piecewise constant and piecewise affine in-

terpolants of the values {z(kτ)}T/τk=1. Alternatively, without such interpolants, one

can use an analog of (11.109) as∫ T

0

⟨A(duτ

dt

), z⟩+⟨B(uτ ), z

⟩− ⟨fτ , z⟩− ⟨[duτ

dt

]i,dz

dt

⟩dt

+⟨duτ

dt(T ), z(T )

⟩=⟨v0, z(0)

⟩(11.128)

and prove the convergence of the piece-wise affine interpolant [duτ

dt ]i towards dudt .

11.4 Exercises

Exercise 11.38 (Penalty-function method for type-II parabolic inequalities). Con-sider the complementarity problem (11.25). Show the connection between (11.25)and (11.26) analogously as done in Proposition 5.9. The L2-type penalty-functionmethod leads to the initial-boundary-value problem

∂uε

∂t+

1

ε

(∂uε

∂t

)−−Δuε + c(uε) = g in Q,

uε = 0 on Σ,

uε(0, ·) = u0 on Ω.

⎫⎪⎪⎬⎪⎪⎭ (11.129)


As (11.37) above, test (11.129) by ∂∂tuε and formulate assumptions on u0 and on

c(·) to obtain the estimates∥∥uε

∥∥L∞(I;W 1,2

0 (Ω))≤ C,

∥∥∥∂uε

∂t

∥∥∥L2(Q)

≤ C,∥∥∥(∂uε

∂t

)−∥∥∥L2(Q)

≤ C√ε. (11.130)

Test (11.129) by v − ∂∂tuε and show the convergence of a selected subsequence

{uε}ε>0 to the solution of (11.26).31

Exercise 11.39 (Landau-Lifschitz-Gilbert equation modified). Consider the modi-fication of (11.89):

∂ψ(∂u∂t

)− β(|u|)u× ∂u

∂t− μΔu+ c(u) = g (11.131)

with ψ : R3 → R a convex (possibly nonsmooth) function; this modifica-tion describes pinning effects of dry-friction type [374]. Using the orthogonality(β(|u|)u×∂u

∂t )·∂u∂t = 0, one can design the definition of the weak solution in thespirit of (11.81) as∫

Q

ψ(v) + μ∇u:∇(v−∂u

∂t

)+(c(u)−g

)(v−∂u

∂t

)+(β(|u|)u×∂u

∂t

)·v dxdt ≥

∫Q

ψ(∂u∂t

)dxdt. (11.132)

31Hint: For v ≥ 0, it holds that∫Q

(∂uε

∂t+c(uε)−g

)(v−∂uε

∂t

)+∇uε · ∇

(v−∂uε

∂t

)dxdt =

1

ε

∫Q

(∂uε

∂t

)−(∂uε

∂t−v

)dxdt ≥ 0

so that, formally, we have

0 ≤ lim supε→0

∫Q

(∂uε

∂t+c(uε)−g

)(v−∂uε

∂t

)+∇uε · ∇

(v−∂uε

∂t

)dxdt

= − lim infε→0

(∥∥∥∂uε

∂t

∥∥∥2

L2(Q)+

1

2

∥∥∇uε(T, ·)∥∥2L2(Ω;Rn)

)

+ limε→0

∫Q

(c(uε)−g

)(v−∂uε

∂t

)+

∂uε

∂tv +∇uε · ∇vdxdt +

1

2

∥∥∇u0

∥∥2L2(Ω;Rn)

≤∫Q

(∂u

∂t+c(u)−g

)(v−∂u

∂t

)+∇uε · ∇

(v−∂u

∂t

)dxdt

where also∇uε(T, ·) ⇀ ∇u(T, ·) weakly in L2(Ω;Rn) has been used. Note that, as we do not have∇ ∂

∂tuε ∈ L2(Q) guaranteed by our a-priori estimates (11.130), the term

∫Q∇uε ·∇ ∂

∂tuεdxdt gets

a meaning only if put equal to 12‖∇uε(T, ·)‖2L2(Ω;Rn)

− 12‖u0‖2L2(Ω;Rn)

, which can be justified

either by using Galerkin’s approximation or by a limit of mollified uε.Finally, as ξ �→ ‖ξ−‖2

L2(Q)is a convex continuous functional on L2(Q), it is weakly lower

semicontinuous, and by ∂uε/∂t ⇀ ∂u/∂t and by the last estimate in (11.130) we have∥∥∥(∂u

∂t

)−∥∥∥2L2(Q)

≤ lim infε→0

∥∥∥(∂uε

∂t

)−∥∥∥2L2(Q)

≤ limε→0

C2ε = 0

so that ∂∂t

u ≥ 0 a.e. in Q.

11.4. Exercises 387

Modify the semi-implicit discretisation (11.90) and prove existence of its weaksolution, show a-priori estimates and make a limit passage in a discrete analog of(11.132).

Exercise 11.40. Consider the initial-boundary-value problem:

∂2u

∂t2−Δ

∂u

∂t+∣∣∣∂u∂t

∣∣∣p−2 ∂u

∂t− div

(|∇u|q−2∇u)= g in Q,

u(0, ·) = u0 ,∂u

∂t(0, ·) = v0, in Ω,

u|Σ = 0 on Σ.

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ (11.133)

Apply the Galerkin method, denote the approximate solution by uk. Qualify u0,v0 and g appropriately and prove a-priori estimates of uk in L∞(I;W 1,q(Ω)) ∩W 1,2(I;W 1,2(Ω)∩Lp(Ω))∩W 1,∞(I;L2(Ω))∩W 2,2(I;W−1,2(Ω)).32 Assume p < q∗

and prove convergence by using monotonicity and the Minty trick.33

32Hint: Test the equation in (11.133) by ∂∂t

uk, obtaining∫Ω

1

2

∂

∂t

∣∣∣∂uk

∂t

∣∣∣2 +∣∣∣∇∂uk

∂t

∣∣∣2 +∣∣∣∂uk

∂t

∣∣∣p +1

q

∂

∂t

∣∣∇uk

∣∣qdx=

∫Ωg(t, ·)∂uk

∂tdx ≤ Cε‖g(t, ·)‖2W1,2(Ω)∗ + ε

∥∥∥∂uk

∂t

∥∥∥2W1,2(Ω)

from which the estimate follows by Gronwall’s inequality assuming u0 ∈ W 1,q(Ω), v0 ∈ L2(Ω),

and g ∈ L2(I;L2∗ (Ω)). For the “dual” estimate of ∂2

∂t2uk use the strategy (11.101).

33Hint: Use monotonicity of the q-Laplacean and (11.133) to write

0 ≤∫Q|∇uk|q−2∇uk − |∇z|q−2∇z · ∇(uk − z) dxdt

=

∫Q

(g−∂2uk

∂t2−∣∣∣∂uk

∂t

∣∣∣p−2 ∂uk

∂t

)(uk−z)−

(∇∂uk

∂t+|∇z|q−2∇z

)· ∇(uk−z) dxdt.

Realize that, by Aubin-Lions’ lemma, ∂∂t

uk → ∂∂t

u (as a subsequence) strongly in L2(Q) due to

the compact embedding L2(I;W 1,2(Ω)) ∩W 1,2(I;W−1,2(Ω)) � L2(Q). Also uk(T, ·) ⇀ u(T, ·)weakly in W 1,2(Ω), hence strongly in L2(Ω), and ∂

∂tuk(T, ·) ⇀ ∂

∂tu(T, ·) weakly in L2(Ω). Then

estimate the limit superior:

lim supk→∞

(−

∫Q

∂2uk

∂t2uk +∇∂uk

∂t· ∇ukdxdt

)= lim sup

k→∞

(∫Q

∣∣∣∂uk

∂t

∣∣∣2dxdt+

∫Ωv0u0 − ∂uk

∂t(T, ·)uk(T, ·) +

1

2

∣∣u0

∣∣2 − 1

2

∣∣∇uk(T, ·)∣∣2dx)

≤∫Q

∣∣∣∂u∂t

∣∣∣2dxdt+ ∫Ωv0u0 − ∂u

∂t(T, ·)u(T, ·) + 1

2

∣∣u0

∣∣2 − 1

2

∣∣∇u(T, ·)∣∣2dx= −

∫ T

0

(⟨∂2u

∂t2, u

⟩+

∫Ω∇∂u

∂t· ∇udx

)dt.

The limit passage in the lower-order term | ∂∂t

uk|p−2 ∂∂t

uk can be made by compactness. If p ≤2, then ∂

∂tuk → ∂

∂tu in Lp(Q) while, if p > 2, then at least Lp−ε(Q) because { ∂

∂tuk}k∈N

is bounded in Lp(Q). If p < q∗, make the limit passage in∫Q | ∂∂tuk|p−2( ∂

∂tuk)ukdxdt when


Exercise 11.41 (Klein-Gordon equation, generalized34). Consider the initial-Dirichlet-boundary-value problem for the semilinear hyperbolic equation

∂2u

∂t2−Δu+ |u|q−2u = g, u(0, ·) = u0,

∂u

∂t(0, ·) = v0, u|Σ = 0. (11.134)

The variant q = 3 is called the Klein-Gordon equation, having applications inquantum physics. For q>1, derive a-priori estimates of u in W 1,∞(I;L2(Ω)) ∩L∞(I;W 1,2

0 (Ω) ∩ Lq(Ω)) by testing it by ∂∂tu. Prove convergence of the Galerkin

approximations uk by weak continuity.35 If q ≥ 2 is small enough, prove uniquenessby using the test function v = ∂

∂tu1−u2, with u1, u2 being two weak solutions.36

Exercise 11.42 (Viscous regularization of Klein-Gordon equation). Consider theinitial-boundary-value problem:

∂2u

∂t2− μdiv

(∣∣∂u∂t

∣∣p−2∇∂u

∂t

)−Δu+ c(u) = g in Q,

u(0, ·) = u0 ,∂u

∂t(0, ·) = v0, in Ω,

u|Σ = 0 on Σ,

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭(11.135)

with μ > 0. Apply the Galerkin method, denote the approximate solution byuk, and prove a-priori estimates for uk in L∞(I;W 1,2(Ω)) ∩ W 1,p(I;W 1,p(Ω)) ∩W 2,p′

(I;Wmax(2,p)(Ω)) and specify qualifications on u0, v0, g, and c(·).37 Prove

realizing boundedness of {uk}k∈N in L∞(I;W 1,q(Ω)) ⊂ Lq∗ (Q). Finally, put z = u + δw andfinish the proof by Minty’s trick.

34Cf. Barbu [38, Sect.4.3.5], Jerome [215], or Lions [261, Sect.I.1].35Hint: For q < p∗ +1, use the Aubin-Lions Lemma 7.7 to get compactness in Lq−1(Q) which

allows for a limit passage through the term∫Q|u|q−2uv dxdt if v ∈ L∞(Q). For q ≥ p∗ + 1,

interpolate between W 1,2(Ω) and Lq(Ω) and get again compactness in Lq−1(Q) but now byLemma 7.8.

36Hint: Abbreviating u12 = u1−u2, realize that r �→ |r|q−2r is Lipschitz continuous on [r1, r2]with the Lipschitz constant (q−1)max(|r1|q−2, |r2|q−2); this test gives

1

2

d

dt

(∥∥∥∂u12

∂t

∥∥∥2L2(Ω)

+∥∥∇u12

∥∥2

L2(Ω;Rn)

)=

∫Ω

(|u2|q−1u2 − |u1|q−1u1

)∂u12

∂tdx

≤ q−12

max(∥∥ |u1|q−2

∥∥Lα(Ω)

,∥∥ |u2|q−2

∥∥Lα(Ω)

)(∥∥u12

∥∥2L2∗ (Ω)

+∥∥∥∂u12

∂t

∥∥∥2L2(Ω)

)with α so that α−1 + (2∗)−1 + 2−1 = 1. Exploiting that u1, u2 ∈ L∞(I;L2∗ (Ω)) and assumingq so small that α ≥ 2∗/(q − 2), proceed by Gronwall inequality.

37Hint: Test the equation in (11.135) by ∂∂t

uk, obtaining∫Ω

1

2

∂

∂t

∣∣∣∂uk

∂t

∣∣∣2 + μ∣∣∣∇∂uk

∂t

∣∣∣p +1

2

∂

∂t

∣∣∇uk

∣∣2dx=

∫Ω

(g(t, ·)− c(uk)

)∂uk

∂tdx ≤ Cε

∥∥g(t, ·)− c(uk)∥∥p′W1,p(Ω)∗ + ε

∥∥∥∂uk

∂t

∥∥∥pW1,p(Ω)

,

from which the claimed estimates follow by Gronwall’s inequality if u0 ∈ W 1,2(Ω), v0 ∈ L2(Ω),

g ∈ Lp′(I;Lp∗′(Ω)), and c(·) has an at most 2/p′-growth. For p < 2, an at most linear growth of

11.4. Exercises 389

convergence by monotonicity and the Minty trick.38 Eventually, denoting uμ thesolution to (11.135), prove that uμ approaches the solution to the Klein-Gordonequation (11.134) if c(r) = |r|q−2r when μ → 0.39

Exercise 11.43 (Martensitic transformation in shape-memory alloys40). Consider

∂2u

∂t2− μΔ

∂u

∂t− div

(σ(∇u)

)+ λΔ2u = g in Q,

u(0, ·) = u0 ,∂u

∂t(0, ·) = v0, in Ω,

u = 0 ,∂u

∂ν= 0, on Σ.

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭(11.136)

Assume μ, λ > 0, g ∈ L2(I;L2∗∗′(Ω)), u0 ∈ W 2,2

0 (Ω), v0 ∈ L2(Ω), and at mostlinear growth of σ : Rn → Rn, consider Galerkin’s approximation, and derivethe a-priori estimates in W 1,∞(I;L2(Ω))∩W 1,2(I;W 1,2(Ω))∩L∞(I;W 2,2

0 (Ω)) by

testing (11.136) by ∂∂tu.

41 Then estimate still ∂2

∂t2u in L2(I;W−2,2(Ω)) and proveconvergence of Galerkin’s approximants. Make also the limit passage in (11.136)

c(·) can be allowed if∫Ωc(uk)

∂∂t

ukdx ≤ 12‖c(uk)‖2L2(Ω)

+ 12‖ ∂∂t

uk‖2L2(Ω)is used. Alternatively,

one can impose a condition c(r)r ≥ 0 and estimate c(uk)∂∂t

uk = ∂∂t

∫ uk0

c(ξ)dξ on the left-hand

side. The “dual” estimate of ∂2

∂t2uk then follows by the strategy (11.101).

38Hint: By the monotonicity of the p-Laplacean and by (11.135),

0 ≤∫Qμ(∣∣∇∂uk

∂t

∣∣p−2∇∂uk

∂t− ∣∣∇∂z

∂t

∣∣p−2∇∂z

∂t

)· ∇∂(uk − z)

∂tdxdt

=

∫Q

(g − c(u)− ∂2uk

∂t2

)∂(uk − z)

∂t−

(∇uk +

∣∣∇∂z

∂t

∣∣p−2∇∂z

∂t

)· ∇∂(uk − z)

∂tdxdt.

By using ∂∂t

uk(T, ·) ⇀ ∂∂t

u(T, ·) weakly in L2(Ω) and uk(T, ·) ⇀ u(T, ·) weakly in

W 1,2(Ω), estimate the limit superior as lim supk→∞∫Q− ∂2

∂t2uk

∂∂t

uk − ∇uk·∇ ∂∂t

ukdxdt =

limsupk→∞12

∫Ω|v0|2−| ∂∂tuk(T, ·)|2+

∣∣∇u0

∣∣2−|∇u(T, ·)|2dx ≤ 12

∫Ω|v0|2−

∣∣∣∂u∂t (T, ·)|2+∣∣∇u0

∣∣2−∣∣∇u(T, ·)∣∣2dx = − ∫Q

∂2u∂t2

∂u∂t

+∇u·∇ ∂u∂t

dxdt.39Hint: Realize that ‖uμ‖W1,p(I;W1,p(Ω)) = O(μ−1/p) hence the term

∫Q μ

∣∣ ∂∂t

u∣∣p−2∇ ∂

∂tu ·

∇v dxdt = O(μ1−1/p) with v fixed vanishes for μ↘ 0.40In the vectorial variant, u(t, ·) : Ω→ Rn is the “displacement”, cf. Example 6.7, and (11.136)

describes isothermal vibrations of a “viscous” solid whose stress response σ : Rn×n → Rn×n neednot be monotone and need not have any quasiconvex (cf. Remark 6.5) potential, and which hassome capilarity-like behaviour with λ > 0 possibly small. The multi-well potential of σ maydescribe various phases (called martensite or austenite) in so-called shape-memory alloys andthen (11.136) is a very simple model for a solid-solid phase transformation, cf. [343] for a criticaldiscussion. For mathematical treatment of this capilarity case see e.g. Abeyaratne and Knowles[1] or Hoffmann and Zochowski [206], cf. also Brokate and Sprekels [71, Chap.5].

41Hint: This test gives

1

2

d

dt

(∥∥∥∂u∂t

∥∥∥2L2(Ω)

+ λ∥∥∇2u

∥∥2L2(Ω;Rn×n)

)+ μ

∥∥∥∂∇u

∂t

∥∥∥2L2(Ω;Rn)

=

∫Ωgu− σ(∇u) · ∂∇u

∂tdx

≤ 1

2

∥∥g∥∥2L2∗∗′

(Ω)+

1

2

∥∥u∥∥2L2∗∗ (Ω)

+1

2μ

∥∥σ(∇u)∥∥2

L2(Ω;Rn)+

μ

2

∥∥∥∂∇u

∂t

∥∥∥2L2(Ω;Rn)


with μ ↘ 0, showing existence of a solution to the semilinear hyperbolic equation∂2

∂t2 u− div(σ(∇u)) + λΔ2u = g.42

Exercise 11.44. Modify Exercise 11.42 by replacing the term c(u) by c(∇u) ordiv(a0(u)

)with a0 : R → Rn.

Exercise 11.45. Show that B := −Δp formulated weakly with Dirichlet boundary

conditions, i.e. V = W 1,p0 (Ω), satisfies (11.103) provided p ≥ 2.43

Exercise 11.46 (Rothe method). Applying the Rothe method to the system (9.64),one obtains the system

vkτ − vk−1τ

τ+Ak

τ

(vkτ)+Bk

τ (ukτ ) = fk

τ , u0τ = u0, (11.137a)

ukτ − uk−1

τ

τ− vkτ = 0 , v0τ = v0, (11.137b)

to be solved recurrently for k = 1, . . . , T/τ . Compare (11.137) with (11.125).44

Consider varying time step τk (which might be useful for efficient numerical im-plementation) and modify (11.125) correspondingly and perform an energy-typeestimate by testing (11.137a) by vkτ .

45


Doubly nonlinear problems from Sections 11.1–11.2 have, in concrete cases,been thoroughly exposed in Visintin [418, Sect.III.1] and, under the name pseu-doparabolic equations, in Gajewski, Groger and Zacharias [168, Chap.V].

In particular, the structure in Section 11.1 has been investigated by Colliand Visintin [101, 104]. Also, differently from Remark 11.10, if Φ is smooth, avariational principle can be devised by usuing the functional46

u �→∫ T

0

Ψ(dudt

)+Ψ∗

(f−Φ′(u)

)+⟨f,

du

dt

⟩dt+Φ(u(T )). (11.138)

and continue by estimation of ‖∇u‖2L2(Ω;Rn)

≤ 2‖∇u0‖2L2(Ω;Rn)+2t

∫ t0‖ ∂∂t∇u‖2

L2(Ω;Rn)and by

Gronwall’s inequality.42Hint: Denoting uμ the weak solution to (11.136), realize that ‖ ∂

∂tuμ‖L2(I;W1,2(Ω)) =

O(1/√μ) while the other estimates are independent of μ > 0, so that the term

∫Q μ∇ ∂

∂tuμ ·

∇v dxdt = O(√μ) and vanishes in the weak formulation if μ→ 0.

43Hint: Paraphrase (8.174) to show 〈B′(u)v, v〉 = ∫Ω|∇u|p−2|∇v|2+(p−2)|∇u|p−4(∇v ·∇u)2

and then use Holder’s inequality to show (11.103).44Hint: Substituting vkτ from (11.137b) into (11.137a) gives exactly (11.125a) but (11.137b) is

not considered for k = 0 so the formal value u−1τ from (11.125b) is not needed now.

45Hint: The second-order difference (ukτ−2uk−1

τ +uk−2τ )/τ2 turns into a non-symmetric differ-

ence ukτ/τ

2k − uk−1

τ (τk+τk−1)/(τ2k τk−1) + uk−2

τ /(τkτk−1).46The construction of (11.54) is by Fenchel identity applied to (Ψ,Ψ∗) at the argument

(dudt

, f−Φ′(u)), which yields the equivalence between ∂Ψ(dudt

) = f−Φ′(u) and that the functional∫ T0 Ψ(du

dt)+Ψ∗(f−Φ′(u))−〈du

dt, f−Φ′(u)〉 dt = ∫ T

0 Ψ(dudt

)+Ψ∗(f−Φ′(u)) + 〈dudt

, f−Φ′(u)〉 dt+Φ(u(T )) −Φ(u0) takes its minimal value, i.e. 0.


For a special quadratic Φ arising in plasticity, see [397] while a general Φ wasconsidered in [398] with Ψ homogeneous degree-1 in both cases, i.e. Ψ(av) = aΨ(v)for any a ≥ 0. A more general Ψ was considered in [280]. In fact, a lot of physicalapplications are based on (11.5) with Ψ homogeneous degree-1. This is related toso-called rate-independent systems and requires q = 1 in (11.31a)–(11.31b) so thatthe results presented in Section 11.1.2 do not cover this case. Instead of the systemof two variational inequalities (11.32), a mathematically more suitable definitionof the solution has been proposed by Mielke and Theil [282] and works merelywith energetics of the process u : [0, T ] → V . This process is called an energeticsolution if, besides the initial condition u(0) = u0, it satisfies the stability and theenergy inequality in the sense:

∀ t∈I ∀v∈V : Φ(u(t))− 〈f(t), u(t)〉 ≤ Φ(v) − 〈f(t), v〉+Ψ(u(t)−v

), (11.139a)

∀ t≥s : Φ(u(t))− 〈f(t), u(t)〉+VarΨ(u; s, t)

≤ Φ(u(s))− 〈f(s), u(s)〉 −∫ t

s

⟨∂f∂ϑ

, u(ϑ)⟩dϑ (11.139b)

where VarΨ(u; s, t) denotes the total variation of Ψ along the process u during

the interval [s, t] defined by VarΨ(u; s, t) := sup∑j

i=1 Ψ(u(ti−1)− u(ti)) with thesupremum taken over all j ∈ N and over all partitions of [0, t] in the form 0 = t0<t1< · · · <tj−1<tj = t. Note that this definition does not involve explicitly timederivative d

dtu which indeed need not exist in an conventional sense. Cf. Mielke[278, 279] for a survey of the related theory and applications, and for other conceptsof solution. Sometimes, a generalization for Ψ = Ψ(u, d

dtu) is useful. The specialcase of a homogeneous degree-1 potential Ψ(u, ·) := [δK(u)]

∗ with a convex set

K(u) ⊂ V then leads to the inclusion ∂du/dtΨ(u, ddtu)+Φ′(u) = [∂δK(u)]

−1( ddtu)+

Φ′(u) � f , i.e. ddtu ∈ NK(u)(f−Φ′(u)). Processes u governed by such inclusions

are called the sweeping processes, see e.g. Krejcı at al. [239, 240, 241] and Kunzeand Monteiro Marques [246].

The doubly nonlinear structure in Sections 11.2 first occurred probably inGrange and Mignot [187], and was investigated in particular by Aizicovici andHokkanen [7], Alt and Luckhaus [11] (both even with a possible degeneracy of theparabolic term), DiBenedetto and Showalter [121], Gajewski [164] (with applica-tion to semiconductors), Groger and Necas [192], Otto [319, 320], Showalter [382],and Stefanelli [395]. A thorough exposition is in the monographs by Hokkanen andMorosanu [207, Chap.10], and Hu and Papageorgiou [209, Part II, Sect.II.5].

The 2nd-order evolution has been addressed by Gajewski et al. [168,Chap.VII], Lions [261, Chap.II.6 and III.6], and Zeidler [427, Chap.33 and 56].The structure of Theorem 11.33(i) even with B set-valued, arising from a concreteunilateral problem, has been addressed by Jarusek et al. [213]. For both A and B

potential and nonlinear in the highest derivatives, e.g., ∂2

∂t2u−Δp∂∂tu−Δqu = g,

see Bulıcek, Malek, and Rajagopal [84] who assumed p ≥ 2 and q ≤ 2, improvingthus former results by Friedman and Necas [156]. A similar problem is also inBiazutti [52].

Chapter 12

Systems of equations: particularexamples

Just as in steady-state problems, no abstract theory exists universally for a broaderclass of systems of nonlinear equations.1 Thus, as in Chapter 6, we confine ourselvesto some illustrative examples having straightforward physical motivation and usingthe previously exposed techniques in a nontrivial manner.

12.1 Thermo-visco-elasticity

The first example is indeed rather nontrivial, illustrating on a rather simplethermo-mechanical system how the previous L1-type estimates for the heat equa-tion can be executed jointly with the estimates of the remaining part of the system.

We assume a body occupying the domain Ω ⊂ Rn, n ≤ 3, made from isotropiclinearly-responding elastic and heat conductive material described in terms of thesmall strains. Let us briefly derive a thermodynamically consistent system. Thedeparture point is the specific Helmholtz free energy considered here as:

ψ(e, θ) :=λe

2

(Tr(e)

)2+ μe|e|2− αTr(e)θ − ψ0(θ) (12.1)

and the dissipation rate:2

ξ(e) := λv(Tr(e))2

+ 2μv|e|2, e =∂e

∂t, (12.2)

where (and in following formulae):

1Some of the previous results, however, can be adopted for systems of a special form simplyby considering u vector-valued, see e.g. Ladyzhenskaya et al. [249, Chap.VII].

2Up to the factor 12related to 2-homogeneity, ξ is also a “quasipotential” of dissipative forces.


393


u : Ω → Rn is the displacement, cf. Example 6.7,θ : Ω → R temperature,e = e(u) = 1

2 (∇u) + 12∇u the small-strain tensor, cf. Example 6.8,

μe, λe are Lame constants related to elastic response, cf. (6.23),α= α0(nλe+2μe) with α0 the thermal dilatation coefficient,ψ0(θ) the thermal part of the free energy,c(θ) := θψ′′0 (θ) is heat capacity,� mass density,μv, λv are Lame constants related to viscous response,κ heat conduction coefficient,g the prescribed bulk force,h the prescribed surface force,f the external prescribed heat flux through the boundary Γ.

Of course, “Tr(·)” in (12.1) stands for a trace of a square matrix. The particularterms in (12.1) are related respectively to the elastic stored energy, temperaturedilatation, and a contribution of chaotic vibrations of the atomic grid (=heat).The quadratic form of ξ in (12.2) is related to linear viscosity.

Moreover, as standard in thermodynamics, we define the specific entropy byso-called Gibbs’ relation s = −ψ′θ (≡ − ∂

∂θψ) and the specific internal energy w by

w := ψ + θs =λe

2(div u)2 + μe

∣∣e(u)∣∣2 + ϑ where ϑ = θψ′0(θ)− ψ(θ). (12.3)

The elastic and viscous stress tensors are defined as ψ′e and 12ξ′. The equilibrium

equation balances the total stress σ = ψ′e +12ξ′ with the inertial forces and outer

loading g:

�∂2u

∂t2− div σ = g with σ=div

(λeu+λv

∂u

∂t

)I+2e

(μeu+μv

∂u

∂t

)− αθI, (12.4)

where I∈Rn×n is the identity matrix. The heat equation then can be obtained fromthe energy balance requiring that the kinetic energy and the internal energy in aclosed system is preserved, cf. (12.11) below. Defining still the heat flux −κ(θ)∇θ(isotropic nonlinear medium), we complete (12.4) by the heat equation

θ∂s

∂t= c(θ)

∂θ

∂t+ θ α div

∂u

∂t= div(κ(θ)∇θ) + ξ

(e(∂u∂t

))= div(κ(θ)∇θ) + λv

(div

∂u

∂t

)2+ 2μv

∣∣∣e(∂u∂t

)∣∣∣2 (12.5)

and finally we choose some boundary conditions, e.g. an unsupported mechanicallyand thermally loaded body, and initial conditions:

ν · σ = h, κ(θ)∂θ

∂ν= f on Σ, (12.6a)

u(0, ·) = u0,∂u

∂t(0, ·) = v0, θ(0, ·) = θ0 on Ω. (12.6b)

12.1. Thermo-visco-elasticity 395

The important fact is that the above procedure satisfies the 2nd thermodynamicallaw.3

We employ the enthalpy transformation, cf. Examples 8.71 and 11.27. We

denote ϑ = c (θ) :=∫ θ

0c(�)d� and abbreviate γ(ϑ) := c−1(ϑ) for ϑ ≥ 0 and,

formally, γ(ϑ) = 0 for ϑ < 0. As in Example 8.71, we define β := κ◦γ with κ aprimitive of κ. Note that ϑ has the same meaning as in (12.3) up to the constantψ0(0). By such substitution, the resulted system then takes the form

�∂2u

∂t2− div

(div(λeu+λv

∂u

∂t

)I+2e

(μeu+μv

∂u

∂t

)− αγ(ϑ)I)= g, (12.7a)

∂ϑ

∂t−Δβ(ϑ) = λv

(div

∂u

∂t

)2+ 2μv

∣∣∣e(∂u∂t

)∣∣∣2 − αγ(ϑ)div∂u

∂t, (12.7b)

with the boundary and initial conditions(div(λeu+λv

∂u

∂t

)I+2e

(μeu+μv

∂u

∂t

)− αγ(ϑ)I)· ν = h,

∂β(ϑ)

∂ν= f

⎫⎪⎬⎪⎭ on Σ, (12.8a)

u(0, ·) = u0,∂u

∂t(0, ·) = v0, ϑ(0, ·) = γ−1(θ0) on Ω. (12.8b)

The energy balance can be obtained formally by multiplication of (12.4) by∂∂tu and (12.5) by 1, and by using Green’s formula for (12.4) and (12.5):

d

dt

(�2

∥∥∥∂u∂t

∥∥∥2L2(Ω)

+

∫Ω

ϕ(e(u)) dx)

+

∫Ω

(ξ(e(∂u

∂t))− αγ(ϑ)div

∂u

∂t

)dx =

∫Ω

g·∂u∂t

dx+

∫Γ

h·∂u∂t

dS, (12.9)

d

dt

∫Ω

ϑdx−∫Ω

(ξ(e(∂u

∂t))− αγ(ϑ)div

∂u

∂t

)dx =

∫Γ

f dS, (12.10)

where ϕ(e) := ψ(e, 0). Summing (12.9) with (12.10), we get the total-energy bal-ance:

d

dt

(�2

∥∥∥∂u∂t

∥∥∥2L2(Ω)︸︷︷︸

kinetic energy

+

∫Ω

w(t, x) dx︸︷︷︸internal energy

):=

d

dt

∫Ω

�

2

∣∣∣∂u∂t

∣∣∣2+ ϕ(e(u)) + ϑ dx

=

∫Ω

g·∂u∂t

dx +

∫Γ

h·∂u∂t

dS︸︷︷︸power of external forces

+

∫Γ

f dS︸︷︷︸power of exter-nal heat flux

. (12.11)

3Indeed, dividing (12.5) by θ, the Clausius-Duhem inequality reads as:

d

dt

∫Ωs(t, x) dx =

∫Ω

ξ(e(∂u

∂t))+div(κ(θ)∇θ)

θdx =

∫Ω

ξ(e( ∂∂t

u))

θ+ κ(θ)

|∇θ|2θ2

dx+

∫Γ

f

θdS ≥ 0

provided θ > 0 and f ≥ 0. For the positivity of temperature θ, cf. Remark 12.10 below.


To prove existence of a weak solution of this system, we apply the Rothemethod with a suitable regularization to compensate the growth of the non-monotone terms on the right-hand side of the heat equation (12.7b). More specif-ically, we consider

�ukτ−2uk−1

τ +uk−2τ

τ2− div

(div(λeu

kτ+λv

ukτ−uk−1

τ

τ

)I− αγ(ϑk

τ )I

+2e(μeu

kτ+μv

ukτ−uk−1

τ

τ

))= gkτ + τdiv

(|e(ukτ )|η−2e(uk

τ )), (12.12a)

ϑkτ−ϑk−1

τ

τ−Δβ(ϑk

τ ) = λv

(div

ukτ−uk−1

τ

τ

)2+ 2μv

∣∣∣e(ukτ−uk−1

τ

τ

)∣∣∣2 − αγ(ϑkτ )div

ukτ−uk−1

τ

τ, (12.12b)

with the corresponding boundary and the regularized initial conditions(div(λeu

kτ+λv

ukτ−uk−1

τ

τ

)I− αγ(ϑk

τ )I+ τ |e(ukτ )|η−2e(uk

τ )

+2e(μeu

kτ+μv

ukτ−uk−1

τ

τ

))· ν = 0,

∂β(ϑkτ )

∂ν= fk

τ on Γ, (12.13a)

u0τ = u0τ , u−1

τ = u0τ − τv0, ϑ0

τ = γ−1(θ0) on Ω. (12.13b)

In terms of the original data, we assume:

∃cmin>0, ω > 2n/(n+2) ∀θ∈R+ : c(θ) ≥ cmin(1+θ)ω−1, (12.14a)

∃β0 > 0 ∀θ∈R+ : κ(θ) ≥ β0c(θ), (12.14b)

� > 0, μe > 0, μv > 0, λe > −2μe/n, λv > −2μv/n, (12.14c)

g ∈ L2(I;L2∗′(Ω;Rn)), h ∈ L2(I;L2#

′(Γ;Rn)), (12.14d)

u0 ∈ W 1,2(Ω;Rn), v0 ∈ L2(Ω;Rn), (12.14e)

θ0 ≥ 0, f ≥ 0, c (θ0) ∈ L1(Ω), f ∈ L1(Σ). (12.14f)

Note that (12.14a) requires a slight growth of the heat capacity and conductivityif n ≥ 2 and that it bounds the growth of γ as |γ(ϑ)| ≤ C(1+ϑ1/ω). Also notethat, due to (12.14b), it holds that β′(ϑ) = [κ/c](γ(ϑ)) ≥ β0.

Lemma 12.1 (Existence of Rothe’s solution). Let (12.14) hold and η > 4, andlet u0τ ∈ W 1,η(Ω;Rn). Then the boundary-value problem (12.12)–(12.13) has asolution (uk

τ , ϑkτ ) ∈ W 1,η(Ω;Rn)×W 1,2(Ω) such that ϑk

τ ≥ 0 for any k = 1, ..., T/τ .

Lemma 12.2 (Energy estimates). Let (12.14) hold and η > 4, and let also‖u0τ‖W 1,η(Ω;Rn) = O(τ−1/η). Then:


∥∥uτ

∥∥W 1,∞(I;L2(Ω;Rn))∩L∞(I;W 1,2(Ω;Rn))

≤ C, (12.15a)∥∥ϑτ

∥∥L∞(I;L1(Ω))

≤ C, (12.15b)∥∥e(uτ )∥∥L∞(I;Lη(Ω;Rn×n))

≤ Cτ−1/η. (12.15c)

Proposition 12.3 (Further estimates). Under the assumption of Lemma 12.2, italso holds that∥∥uτ

∥∥W 1,2(I;W 1,2(Ω;Rn))

≤ C, (12.16a)∥∥∇ϑτ

∥∥Lr(Q;Rn)

≤ Cr with r <n+2

n+1, (12.16b)∥∥ϑτ

∥∥Lq(Q)

≤ Cq with q <n+2

n, (12.16c)∥∥∥ ∂

∂t

[∂uτ

∂t

]i∥∥∥L2(I;W 1,2(Ω;Rn)∗)+Lη′(I;W 1,η(Ω;Rn)∗)

≤ C, (12.16d)∥∥∥ ∂

∂t

[∂uτ

∂t

]i − τdiv(|e(uτ )|η−2e(uτ )

)∥∥∥L2(I;W 1,2(Ω;Rn)∗)

≤ C, (12.16e)∥∥∥∂ϑτ

∂t

∥∥∥L1(I;W 3,2(Ω)∗)

≤ C. (12.16f)

Proof. We use the test of (12.12b) by χ(ϑkτ ) := 1 − (1+ϑk

τ )−ε as suggested in

Remark 9.25 for p = 2. After summation for k = 1, ..., T/τ , we obtain

L := β0ε

∫Q

|∇ϑτ |2(1+ϑτ )1+ε

dxdt = β0

∫Q

χ′(ϑτ )|∇ϑτ |2 dxdt

≤∫Q

χ′(ϑτ )β′(ϑτ )∇ϑτ ·∇ϑτ dxdt =

∫Q

∇β(ϑτ )·∇χ(ϑτ ) dxdt

≤∫Q

∇β(ϑτ )·∇χ(ϑτ ) dxdt+

∫Ω

χ(ϑτ (T, ·)) dx

≤∫Ω

χ(ϑ0) dx+

∫Q

rτχ(ϑτ ) dxdt+

∫Σ

fτχ(ϑτ ) dSdt

≤ ∥∥γ(θ0)∥∥L1(Ω)+∥∥rτ∥∥L1(Q)

+∥∥fτ∥∥L1(Σ)

=: C0 +∥∥rτ∥∥L1(Q)

, (12.17)

where χ is a primitive of χ such that χ(0) = 0 and where we abbreviated theheat sources rτ := ξ(e( ∂

∂tuτ ))− αγ(ϑτ )div∂∂tuτ . The inequality on the 4th line of

(12.17) uses monotonicity of χ hence convexity of χ so that the “discrete chainrule” holds:

χ(ϑkτ )− χ(ϑk−1

τ )

τ≤ ϑk

τ − ϑk−1τ

τχ(ϑk

τ ).

Further, we test (12.12b) by ukτ−uk−1

τ , sum it for k = 1, . . . , T/τ , and add to(12.17) with a sufficiently big weight to see the dissipation ξ(e( ∂

∂tuτ )) on the left-


hand side. Thus, we obtain

ξ(e(∂uτ

∂t

))+ L ≤ C1 + C1

∥∥∥αγ(ϑτ )div∂uτ

∂t

∥∥∥L1(Q)

(12.18)

with some C1. By calculations like4 (9.53) with p = 2 and q = 1−ε, for 1 ≤ r < 2,we obtain∫

Q

∣∣∇ϑτ

∣∣rdxdt ≤ ( L

β0ε

)r/2( ∫ T

0

∥∥1+ϑτ (t, ·)∥∥(1+ε)r/(2−r)

L(1+ε)r/(2−r)(Ω)dt)(2−r)/2

. (12.19)

Then we use (9.54) to obtain∥∥1+ϑτ (t, ·)∥∥L(1+ε)r/(2−r)(Ω)

≤ CGN

(∥∥1+ϑτ (t, ·)∥∥L1(Ω)

+∥∥∇ϑτ (t, ·)

∥∥Lr(Ω;Rn)

)λ∥∥1+ϑτ (t, ·)∥∥1−λ

L1(Ω)

≤ CGN

(measn(Ω)+C

)1−λ(measn(Ω) + C +

∥∥∇ϑτ (t, ·)∥∥Lr(Ω;Rn)

)λ(12.20)

with C from (12.15b), provided (9.55) holds, i.e. provided

2− r

(1 + ε)r≥ λ

(1r− 1

n

)+ 1− λ with 0 < λ ≤ 1. (12.21)

We raise (12.20) to the power (1+ε)r/(2−r), use it in (12.19), and choose λ :=(2−r)/(1+ε). As in (9.56), we obtain

( ∫ T

0

∥∥1+ϑτ (t, ·)∥∥(1+ε)r/(2−r)

L(1+ε)r/(2−r)(Ω)dt) 2−r

2 ≤ C2 + C2

(∫Q

|∇ϑτ |rdxdt) 2−r

2

. (12.22)

Merging (12.19) with (12.17) and with (12.22) gives an estimate of the type

‖∇ϑτ‖rLr(Q;Rn)/(1 + ‖∇ϑτ‖r(1−r/2)Lr(Q;Rn)) ≤ C(1 + ‖rτ‖L1(Q))

r/2, i.e.,

‖∇ϑτ‖rLr(Q;Rn) − C3 ≤( ‖∇ϑτ‖rLr(Q;Rn)

1 + ‖∇ϑτ‖r(1−r/2)Lr(Q;Rn)

)2/r

≤ C3

(1 + ‖rτ‖L1(Q)

)(12.23)

for C3 large enough. Substituting our choice of λ := (2−r)/(1+ε) into (12.21)5,one gets, after some algebra, the condition

r ≤ 2 + n− εn

1 + n. (12.24)

4We use 1+ϑτ instead of u in (9.53). In fact, C1/(q−1) in (9.53) is now L/(β0ε) so that wecan indeed consider q < 1.

5Note that 0 < λ < 1 needed in (12.21) is automatically ensured by 1 ≤ r < 2 and ε > 0.


For 0 < ω ≤ 2, using that γ(·) has a growth at most 1/ω, we can estimatethe last term in (12.18) as∣∣∣ ∫

Q

αγ(ϑτ )div∂uτ

∂tdxdt

∣∣∣ ≤ C∥∥γ(ϑτ )

∥∥2L2(Q)

+ δ∥∥∥∂e(uτ )

∂t

∥∥∥2L2(Q;Rn×n)

≤ Cδ + Cδ

∥∥ϑτ

∥∥2/ωL2/ω(Q)

+ δ∥∥∥∂e(uτ )

∂t

∥∥∥2L2(Q;Rn×n)

(12.25)

and absorb the last term in the left-hand side of (12.18) by choosing δ < 2C2ε1.For ω ≥ 2, we can simply make this estimate like in the case ω = 2. Further, bythe Gagliardo-Nirenberg inequality, we estimate:

∥∥ϑτ

∥∥L2/ω(Ω)

≤ KGN

∥∥ϑτ

∥∥1−μ

L1(Ω)

(∥∥ϑτ

∥∥L1(Ω)

+∥∥∇ϑτ

∥∥Lr(Ω;Rn)

)μ(12.26)

for

ω

2≥ μ

(1r− 1

n

)+ 1− μ, 0 ≤ μ < 1, (12.27)

and, by (12.19), (12.22), and the first estimate in (12.23), we have ‖∇ϑτ‖Lr(Ω;Rd) ≤C3 + C

2/r2 L so that we can estimate

Cδ

∥∥ϑτ

∥∥2/ωL2/ω(Ω)

≤ CδK2/ωGN

C2(1−μ)/ω3

(C3 +

∥∥∇ϑτ

∥∥Lr(Ω;Rn)

)2μ/ω≤ C′δ + δ

∥∥∇ϑτ

∥∥rLr(Ω;Rn)

≤ C′δ + δC3 + δC2/r2 L (12.28)

provided 2μ/ω < r, i.e.,

ω

2>

μ

r. (12.29)

The last term in (12.28) is to be absorbed in the left-hand side of (12.18). Theoptimal choice of μ makes the right-hand sides of (12.27) and (12.29) mutuallyequal, which gives μ = n/(n+1). Note that always 0 < μ < 1, as required for(12.26). Taking into account that r < (n+2)/(n+1), from (12.27) (or, equally,from (12.29)) we obtain

ω >2n

n+ 2. (12.30)

It eventually gives the estimates (12.16a) and (12.16b).

The estimate (12.16c) arises by a suitable interpolation between (12.15b)and (12.16b). The remaining “dual” estimates in (12.16) follows from the alreadyobtained ones. �


Proposition 12.4 (Convergence for τ → 0). Let (12.14) hold and η > 4, and letalso u0τ → u0 in W 1,2(Ω;Rn) and even ‖u0τ‖W 1,η(Ω;Rn) = o(τ−1/η). Then thereis a subsequence such that

uτ → u strongly in W 1,2(I;W 1,2(Ω;Rn)), (12.31a)

ϑτ → ϑ strongly in Lq(Q) with any 1 ≤ q < (n+2)/n, (12.31b)

and any (u, ϑ) obtained by this way is a weak solution to the initial-boundary-valueproblem (12.7)–(12.8). In particular, (12.7)–(12.8) has a weak solution.

Proof. Choose a weakly converging subsequence in the topology of the esti-mates (12.16). By the interpolated Aubin-Lions’ Lemma 7.8, combining (12.15b),(12.16b) and (12.16f), one obtains (12.31b).

Like in (11.109) with Remark 11.37, we use the by-part summation (11.126)to obtain the identity∫

Ω

�∂uτ

∂t(T )·vτ (T ) dx−

∫ T

τ

∫Ω

�∂uτ

∂t(· − τ)·∂vτ

∂tdxdt

+

∫Q

div(λeuτ + λv

∂uτ

∂t− αγ(ϑτ )I

)div vτ

+ 2e(μeuτ+μv

∂uτ

∂t+ τ |e(uτ )|η−2e(uτ )

):e(vτ ) dxdt

=

∫Ω

v0·vτ (τ) dx +

∫Q

gτ ·vτ dxdt+∫Σ

hτ ·vτ dxdt, (12.32)

where vτ and vτ is the piecewise constant and the affine interpolants of the test

function values {v(kτ)}T/τk=1.

For v smooth, by (12.15c), one has∣∣∣ ∫Q

τ |e(uτ )|η−2e(uτ ):e(v) dxdt∣∣∣ ≤ τ

∥∥e(uτ )∥∥η−1

Lη(Q;Rn×n)

∥∥e(v)∥∥Lη(Q;Rn×n)

= O(τ1/η)

(12.33)

and thus can see that the regularizing η-term disappears in the limit for τ → 0.Altogether, we proved the convergence of (12.32) to the weak formulation of themechanical part, namely∫

Ω

�∂u

∂t(T )·v(T ) dx−

∫Q

�∂u

∂t·∂v∂t

+ div(λeu+ λv

∂u

∂t− αγ(ϑ)I

)div v

+ 2e(μeuτ+μv

∂u

∂t

):e(v) dxdt =

∫Ω

v0·v(0) dx+

∫Q

g·v dxdt+∫Σ

h·v dxdt. (12.34)

By (12.16e), it holds that

�∂

∂t

[∂uτ

∂t

]i− τdiv(|e(uτ )|η−2e(uτ )

)→ ζ weakly in L2(I;W 1,2(Ω;Rn)∗) (12.35)


and, like in Exercise 8.85, we can show that ζ = �∂2u∂t2 .

It is essential that �∂2u∂t2 is in duality with ∂u

∂t , so that we can legally substitute

v = ∂u∂t into (12.34) to obtain mechanical-energy equality∫

Ω

�

2

∣∣∣∂u∂t

(T )∣∣∣2 + ϕ

(e(u(T ))

)dx+

∫Q

ξ(e(∂u∂t

))dxdt =

∫Ω

�

2|v0|2

+ϕ(e(u0)) dx+

∫Q

g·∂u∂t

+ αγ(ϑ)div∂u

∂tdxdt+

∫Σ

h·∂u∂t

dSdt (12.36)

with ϕ(e) = ψ(e, 0) as in (12.9).

For the limit passage in the heat equation, we need to prove strong conver-gence of e( ∂

∂tuτ ) in L2(Q;Rn×n). We use∫Q

ξ(e(∂u∂t

))dxdt ≤ lim inf

τ→0

∫Q

ξ(e(∂uτ

∂t

))dxdt ≤ lim sup

τ→0

∫Q

ξ(e(∂uτ

∂t

))dxdt

≤ lim supτ→0

∫Ω

�

2|v0|2 − �

2

∣∣∣∂uτ

∂t(T )

∣∣∣2 + ϕ(e(u0τ ))− ϕ(e(uτ (T ))

)+

τ

η

∣∣e(u0τ )∣∣η

− τ

η

∣∣e(uτ(T ))∣∣ηdx +

∫Q

g·∂uτ

∂t+αγ(ϑτ )div

∂uτ

∂tdxdt+

∫Σ

h·∂uτ

∂tdSdt

≤∫Ω

�

2|v0|2 − �

2

∣∣∣∂u∂t

(T )∣∣∣2 + ϕ(e(u0))− ϕ

(e(u(T ))

)dx

+

∫Q

g·∂u∂t

+ αγ(ϑ)div∂u

∂tdxdt+

∫Σ

h·∂u∂t

dSdt =

∫Q

ξ(e(∂u∂t

))dxdt.

Note that the last equality is exactly (12.36). Thus limτ→0

∫Qξ(e( ∂

∂tuτ )) dxdt =∫Q ξ(e( ∂

∂tu)) dxdt. By (12.14c), the quadratic form ξ is coercive and thus

(∫Qξ(e(·)) dxdt)1/2 is an equivalent norm on L2(Q;Rn×n

sym ) which keeps it uni-

formly convex. Therefore e( ∂∂tuτ ) → e( ∂

∂tu) strongly in L2(Q;Rn×n). Then thelimit passage in the semi-linear heat equation is simple.

In particular, using Lemma 12.1 and the above arguments, some weak solu-tion to (12.7)–(12.8) indeed exists because, due to the qualification (12.14e) of u0,the regularization u0τ with all above required properties always exists. �

Exercise 12.5. Prove Lemma 12.1 by using coercivity6 and pseudomonotonicityof the underlying mapping and then using Brezis’ Theorem 2.6. Further provenon-negativity7 of temperature, i.e. ϑk

τ ≥ 0.

6Hint: test (12.12a) by ukτ and (12.12b) by ϑk

τ and perform the a-priori estimate. Estimatethe nonmonotone terms as

∫Ω|e(uk

τ )|2ϑkτdx ≤ Cε,η+ε‖e(uk

τ )‖ηLη(Ω;Rn×n)+ε‖ϑk

τ‖2L2(Ω)provided

η > 0, and also | ∫Ωγ(ϑk

τ )div(ukτ )ϑ

kτdx| ≤ Cε,η+ε‖e(uk

τ )‖ηLη(Ω;Rn×n)+ε‖ϑk

τ‖2L2(Ω)when realizing

that γ(·) as at most linear growth, and similarly for the term∫Ωγ(ϑk

τ )div ukτdx.

7Hint: test (12.12b) by (ϑkτ )

− and realize that γ(ϑ) = 0 for ϑ ≤ 0.


Exercise 12.6. Prove Lemma 12.2 by imitating (12.11).8

Exercise 12.7. Make the interpolation between (12.15b) and (12.16b) in such away so as to obtain (12.16c).

Exercise 12.8. Prove the strong convergence of e( ∂∂tuτ ) in L2(Q;Rn×n) by the

direct estimate of e( ∂∂t (uτ−u)) similar to (8.217), using also Remark 8.11.

Exercise 12.9. Assuming ω > 2 instead of ω > 2n/(n+2) in (12.14a), prove theestimates (12.15) and (12.16a) directly without the interpolation (12.17)–(12.28)by making the test of (12.12a) by (uk

τ−uk−1τ )/τ and of (12.12b) by 1/2. 9

Remark 12.10 (Positivity of temperature). It is a rather delicate interplay betweenpossible cooling via the adiabatic term and sufficient heating via the dissipativeterm which guarantees positivity of temperature in thermally coupled systems, cf.[102, 146]. Here one can adapt the procedure from Remark 9.28. One can estimate(12.7b) as

∂ϑ

∂t−Δβ(ϑ) = λv

(div

∂u

∂t

)2+ 2μv

∣∣∣e(∂u∂t

)∣∣∣2 − αγ(ϑ)div∂u

∂t

≥ λv

2

(div

∂u

∂t

)2+ 2μv

∣∣∣e(∂u∂t

)∣∣∣2 − α

2

∣∣∣γ(ϑ)∣∣∣2 ≥ −α

2C|ϑ|2, (12.37)

where C comes from the growth restriction |γ(ϑ)| ≤ Cϑ which is ensuredby (12.14a) and the definition γ = c−1. Assuming ϑ0(·) ≥ ϑ0,min > 0, wecompare (12.37) with a solution to the Riccati ordinary-differential equationddtχ + α

2Cχ2 = 0 which, for χ(0) = ϑ0,min > 0, gives a sub-solution of the heatequation. This initial-value problem has the solution χ(t) = 2/(αCt+2/ϑ0,min) ≥2/(αCT+2/ϑ0,min) > 0. Considering ϑsub(t, x) = χ(t), we can subtract (12.37)from ∂

∂tϑsub+α2Cϑ2

sub = 0, test by (ϑ−ϑsub)− ≤ 0, integrate over Ω at each t ∈ I,

and use Green’s formula. Thus we obtain

1

2

d

dt

∫Ω

((ϑ−ϑsub)

−)2dx ≤∫Ω

(ϑ−ϑsub)− ∂

∂t(ϑ−ϑsub) dx

≤∫Ω

α

2C(ϑ2−ϑ2

sub

)(ϑ−ϑsub

)− − β′(ϑ)∇ϑ·∇(ϑ−ϑsub

)−dx

−∫Γ

f(ϑ−ϑsub

)−dS ≤ 0, (12.38)

where the last inequality is also due to ϑ ≥ 0 proved previously, and also due tothe assumption f ≥ 0 a.e. in Σ, and eventually also due to

β′(ϑ)∇ϑ·∇(ϑ−ϑsub

)−= β′(ϑ)∇(ϑ−ϑsub

)·∇(ϑ−ϑsub

)−= β′(ϑ)∇(ϑ−ϑsub

)−·∇(ϑ−ϑsub

)− ≥ 0 (12.39)

8Hint: test (12.12a) by ukτ−uk−1

τ and (12.12b) by 1 and realize the cancellation effects ofthe heat sources like in (12.11). Use also that ϑτ is bounded from below, as already proved inLemma 12.1.

9Hint: Like (12.25), estimate | ∫Ωαγ(ϑτ )div

∂uτ∂t

dx|≤Cδ+δ‖ϑτ ‖L1(Ω)+δ‖e(∂uτ∂t

)‖2L2(Ω;Rn×n)

.


a.e. on Ω; cf. Proposition 1.28. Realizing the initial condition (ϑ(0)−ϑsub(0))− =

(ϑ0−ϑ0,min(0))− = 0, we easily conclude that (ϑ−ϑsub)

− = 0 a.e. on Q, whence

ϑ(t, x) ≥ ϑsub(t) ≥ 2

αCT+2/ϑ0,min> 0 for a.a. (t, x)∈Q. (12.40)

Remark 12.11 (Galerkin method). Applying the Galerkin method to a thermallycoupled system is not so straightforward because the “nonlinear tests” by ϑ− andby χ(ϑ) (used in Exercise 12.5 and in (12.17), respectively) cannot be executedon finite-dimensional subspaces. Anyhow, careful regularization in the mechanicalpart controlled independently of the Galerkin approximation helps. Here, as in(12.12), one can consider a Galerkin approximation of the system:

�∂2uε

∂t2− div

(div(λeuε+λv

∂uε

∂t

)I+2e

(μeuε+μv

∂uε

∂t

)− αγ(ϑε)I)

= g + ε div(∣∣∣e(∂uε

∂t

)∣∣∣η−2

e(∂uε

∂t

)), (12.41a)

∂ϑε

∂t−Δβ(ϑε) = λv

(div

∂uε

∂t

)2+ 2μv

∣∣∣e(∂uε

∂t

)∣∣∣2 − αγ(ϑε)div∂uε

∂t(12.41b)

with the corresponding boundary conditions with a regularized heat flux fε ∈L2(I;L2#

′(Γ)) and with a regularized initial condition ϑ(0) = ϑ0,ε ∈ W 1,2(Ω).

One then proceeds in the following steps. First, testing the particular equationsin (12.41) by uε and ϑε (meant in its Galerkin approximation), we obtain

d

dt

(�2

∥∥∥∂uε

∂t

∥∥∥2L2(Ω;Rn)

+

∫Ω

ϕ(e(uε)

)dx+

1

2

∥∥ϑε

∥∥2L2(Ω)

)+

∫Ω

ξ(e(∂uε

∂t

))dx+ ε

∥∥∥e(∂uε

∂t

)∥∥∥ηLη(Ω;Rn×n)

+ β0

∥∥∇ϑε

∥∥2L2(Ω;Rn)

=

∫Ω

λv

(div

∂uε

∂t

)2ϑε+ 2μv

∣∣∣e(∂uε

∂t

)∣∣∣2ϑε + αγ(ϑε)(1−ϑε)div∂uε

∂t

+ g·∂uε

∂tdx+

∫Γ

h·∂uε

∂t+ fεϑε dS

≤ Cε,η+ε

2

∥∥∥e(∂uε

∂t

)∥∥∥ηLη(Ω;Rn×n)

+∥∥ϑε

∥∥2L2(Ω)

+

∫Ω

g·∂uε

∂tdx+

∫Γ

h·∂uε

∂t+ fεϑε dS

(12.42)

with Cε,η < +∞ depending on ε and η, and with ϕ(e) = ψ(e, 0) as in (12.9) and ξfrom (12.2). The last inequality in (12.42) holds if η is sufficiently large, namely η >max(4, 2/(1−1/ω)). By the usual Holder/Young-inequality treatment of the lastintegrals in (12.42) and by Gronwall’s inequality, we get estimates of the Galerkinapproximation of uε bounded in W 1,∞(I;L2(Ω;Rn)) ∩W 1,η(I;W 1,η(Ω;Rn)) andof ϑε bounded in L∞(I;L2(Ω)) ∩ L2(I;W 1,2(Ω)). From (12.41b), we then get anestimate of the time derivative of the Galerkin approximation of ϑε. This allows


us to pass to the limit in the Galerkin approximation to obtain a weak solution(uε, ϑε) to (12.41). Then we can perform the “nonlinear tests” of (12.41b) by ϑ−εand by χ(ϑε) to obtain estimates analogous to (12.16), and to converge ε → 0analogously as we did for τ → 0 before.

Let us still recall a variation of this procedure based on a separate Galerkinapproximation of the thermal and the mechanical parts (again combined with someregularization) and make a limit passage successively: first in the heat part, thenexecute the needed nonlinear estimates (and suppress possible regularization), andeventually make a limit passage in the mechanical part, cf. [78, 368].

Remark 12.12 (Semi-implicit Rothe method). The Rothe method (12.12) can bemodified to make decoupling of the problem (like in Remark 8.25 but now littledifferently), which yields a variational structure at each time level (like in Exer-cise 8.74) and which also suggests (after an additional spatial discretisation) anefficient numerical strategy. One can even avoid the regularization by the termτdiv|e(u)|η−2e(u) used in (12.12). Namely, we devise:

�ukτ−2uk−1

τ +uk−2τ

τ2− div

(div(λeu

kτ+λv

ukτ−uk−1

τ

τ

)I

− αγ(ϑk−1τ )I+2e

(μeu

kτ+μv

ukτ−uk−1

τ

τ

))= gkτ (12.43a)

ϑkτ−ϑk−1

τ

τ− div

(β′(ϑk−1

τ )∇ϑkτ

)= λv

(div

ukτ−uk−1

τ

τ

)2+ 2μv

∣∣∣e(ukτ−uk−1

τ

τ

)∣∣∣2 − αγ(ϑkτ )div

ukτ−uk−1

τ

τ, (12.43b)

with corresponding boundary conditions. Like in Exercise 12.5, one can proveϑkτ ≥ 0. Yet, nothing is gratis, and here one must impose a quite strong assumption

on the growth of γ, namely ω > 2 in (12.14a) so that |γ(ϑ)| ≤ Cω(1+ |ϑ|1/2). Onecan then perform a-priori estimates (12.15a,b) and (12.16a) by testing (12.43a) byukτ−uk−1

τ

τ as before in (12.9) but the heat-transfer part (12.43b) is tested by 12 , in-

stead of 1 used previously for (12.10). The adiabatic terms do not cancel, yielding

α(γ(ϑk−1τ )− 1

2γ(ϑkτ ))div

ukτ−uk−1

τ

τ .10 Afterward the a-priori estimate (12.16), simpli-fied by avoiding the regularization, and then convergence can be proved.

Remark 12.13. The thermo-visco-elastic system (12.4)–(12.5) has been treated byDafermos, Hsiao, and Slemrod [113, 115, 388] for n = 1 and constant coefficients c

10This can be estimated by Holder inequality∫Ωα(γ(ϑk−1

τ )− 1

2γ(ϑk

τ ))divukτ−uk−1

τ

τdx ≤ |α|∥∥γ(ϑk−1

τ )− 1

2γ(ϑk

τ )∥∥L2(Ω)

∥∥∥divukτ−uk−1

τ

τ

∥∥∥L2(Ω)

≤ α2C2ω

λv

(measn(Ω) +

∫Ω2ϑk−1

τ + ϑkτ dx

)+

λv

2

∥∥∥divukτ−uk−1

τ

τ

∥∥∥2L2(Ω)

and then treated by the discrete Gronwall inequality.


and κ, while for the multidimensional situation only local-in-time results (togetherwith uniqueness and regularity) have been proved by Bonetti and Bonfanti [61].For c = c(θ) growing like θ1/2+ε, ε > 0, the existence of solutions for n = 3was recently treated by Blanchard and Guibe [54] by a Schauder fixed point,and in [368] by using Galerkin approximation directly for (12.4)–(12.5) withoutenthalpy transformation. Regularity has been treated by Paw�low and Zajaczkowski[328] in case c = c(θ) growing linearly. Considering the heat-transfer coefficient κdependent on ∇θ, the multidimensional case was treated also by Necas at al. [306,309] and by Eck, Jarusek and Krbec [132, Sect.5.4.2.2]. For νv = μv = 0 see Jiangand Racke [217, Chap.7]. For some modified models see e.g. Eck and Jarusek [131].

12.2 Buoyancy-driven viscous flow

The evolution version of the Oberbeck-Boussinesq model from Sect. 6.2 for theNewtonean-fluid case looks as11

∂u

∂t+ (u·∇)u−Δu+∇π = g(1− αθ), (12.44a)

div u = 0, (12.44b)∂θ

∂t+ u · ∇θ − κΔθ = 0, (12.44c)

where the notation is as in Section 6.2; for simplicity, the viscosity coefficient andthe mass density now equals 1. Still we consider the initial conditions and theboundary condition as no-slip for u and as Newton’s condition for θ, i.e.:

u(0, ·) = u0, θ(0, ·) = θ0 on Ω, (12.45a)

u = 0 , κ∂θ

∂ν+ βθ = h on Σ. (12.45b)

As to the data g, h, u0, and θ0, we assume, having in mind n = 3, that

g∈L∞(I;L3(Ω;Rn)), h∈L2(I;L4/3(Γ)

), u0∈L2(Ω;Rn), θ0∈L2(Ω). (12.46)

We are going to use Schauder’s-type fixed-point technique. For (v, ϑ) given,we consider u being the very weak solution to the Oseen equation

∂u

∂t+ (v ·∇)u−Δu+∇π = g(1−αϑ), div u = 0, in Q,

u|Σ = 0 on Σ,

u(0, ·) = u0 on Ω,

⎫⎪⎪⎬⎪⎪⎭ (12.47)

11See, e.g., Lions [261, Ch.I, Sect.9.2] or Straughan [392], or Rajagopal et al. [344]. For a moregeneral model expanding the heat equation by an adiabatic and dissipative heat sources seee.g. [223, 310].


and then θ being the very weak solution to

∂θ

∂t+ v · ∇θ − κΔθ = 0 in Q,

κ∂θ

∂ν+ βθ = h on Σ,

θ(0, ·) = θ0 on Ω.

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (12.48)

Note that, for a given (v, ϑ), (12.47) and (12.48) are linear. We denoteW 1,2

0,div(Ω;Rn) = {v∈W 1,2

0 (Ω;Rn); div v = 0}, cf. (6.29).

Lemma 12.14 (A-priori estimates). Let n ≤ 3 and (12.46) hold. Then there isa very weak solution (u, θ) to (12.47) and (12.48) satisfying, for some C1, . . . , C4,

∥∥u∥∥L2(I;W 1,2

0,div(Ω;Rn))∩L∞(I;L2(Ω;Rn))≤ C1

(1 +

∥∥ϑ∥∥L2(I;W 1,2(Ω))

), (12.49a)∥∥∥∂u

∂t

∥∥∥L4/3(I;W 1,2

0,div(Ω;Rn)∗)≤ C2

(1 +

∥∥ϑ∥∥L2(I;W 1,2(Ω))

+∥∥v∥∥

L2(I;W 1,20,div(Ω;Rn))∩L∞(I;L2(Ω;Rn))

), (12.49b)∥∥θ∥∥

L2(I;W 1,2(Ω))∩L∞(I;L2(Ω))≤ C3, (12.49c)∥∥∥∂θ

∂t

∥∥∥L4/3(I;W−1,2(Ω))

≤ C4

(1 +

∥∥v∥∥L2(I;W 1,2

0,div(Ω;Rn))∩L∞(I;L2(Ω;Rn))

). (12.49d)

Proof. Let us consider the Galerkin approximation of (12.47)–(12.48) and, afterderiving the a-priori estimate, we can pass to the limit by the same strategy as inthe proof of Lemma 12.15 below. We proceed only heuristically.12 Test (12.47) byu and use Green’s theorem:

1

2

d

dt‖u‖2L2(Ω;Rn) +

∫Ω

((v · ∇)u) · u+ |∇u|2 +∇π · u dx

=

∫Ω

g(1− αϑ) · u dx ≤ C‖g‖L3(Ω;Rn)

(1+‖ϑ‖L6(Ω)

)‖u‖L6(Ω;Rn). (12.50)

Using∫Ω((v ·∇)u) ·u dx = 0 provided div v = 0, cf. (6.36), and

∫Ω∇π ·u dx =

− ∫Ωπ div u dx = 0, one obtains (12.49) by Young’s inequality and integration

over I.

Moreover, for v bounded in L2(I;W 1,20,div(Ω;R

n)) ∩ L∞(I;L2(Ω;Rn)), we get

12More precisely, we can perform the estimates (12.50) and (12.53) only in Galerkin’s approx-imations because we do not have the by-part formula at our disposal for very weak solutionsthemselves, and then these bounds are inherited in the limit very weak solution, too.


also the dual estimate (12.49b):13∥∥∥∂u∂t

∥∥∥L4/3(I;W 1,2

0,div(Ω;Rn)∗):= sup

||z||L4(I;W

1,20,div

(Ω;Rn))≤1

⟨∂u∂t

, z⟩

= sup||z||

L4(I;W1,20,div

(Ω;Rn))≤1

∫Q

∇u :∇z + (v ·∇)u·z − g(1− αϑ)z dxdt

≤ ∥∥∇u∥∥L2(Q;Rn×n)

(4√T+N3/2

∥∥v∥∥1/2L2(I;W 1,2(Ω;Rn))

∥∥v∥∥1/2L∞(I;L2(Ω;Rn))

)+NC‖g‖L∞(I;L3(Ω;Rn))

(√T+‖ϑ‖L2(I;L6(Ω))

)(12.51)

where N denotes the norm of the embedding W 1,2(Ω) ⊂ L6(Ω) and where we usedthe Holder inequality and the interpolation (1.63) for the convective term:14∫

Q

(v · ∇)u · z dxdt ≤ ∥∥v∥∥L4(I;L3(Ω;R3))

∥∥∇u∥∥L2(Q;R3×3)

∥∥z∥∥L4(I;L6(Ω;R3))

≤ ∥∥v∥∥1/2L2(I;L6(Ω;R3))

∥∥v∥∥1/2L∞(I;L2(Ω;R3))

∥∥∇u∥∥L2(Q;R3×3)

∥∥z∥∥L4(I;L6(Ω;R3))

. (12.52)

Using (12.49a) and (12.49b), the estimate (12.102c) follows.As to (12.49c), we test (12.48) by θ and use Green’s theorem:

1

2

d

dt‖θ‖2L2(Ω) +

∫Ω

(v · ∇θ)θ dx+ κ

∫Ω

|∇θ|2dx+ β

∫Γ

θ2dS

=

∫Γ

hθ dx ≤ ‖h‖L4/3(Γ)‖θ‖L4(Γ), (12.53)

and then (12.49c) follows by the identity (6.33) and the Poincare inequality (1.56).The estimate (12.49d) follows similarly as (12.51):∥∥∥∂θ

∂t

∥∥∥L4/3(I;W 1,2

0,div(Ω;Rn)∗)≤ ∥∥∇θ

∥∥L2(Q;R3)

((4√T

+N3/2∥∥v∥∥1/2

L2(I;W 1,2(Ω;R3))

∥∥v∥∥1/2L∞(I;L2(Ω;R3))

)+NΓ‖h−βθ‖L2(I;L4(Γ))

)(12.54)

where NΓ denotes the norm of the trace operator W 1,2(Ω) ⊂ L4(Γ). �13Alternatively, with the same effect, we could use Green’s formula and then the Holder in-

equality and interpolation in (12.52):∫Ω(v·∇)u·z dx = −

∫Ω(v·∇)z·u dx ≤ ‖v‖L8/3(I;L4(Ω;R3))‖∇z‖L4(I;L2(Ω;R3×3))‖u‖L8/3(I;L4(Ω;R3))

≤ ‖v‖3/4L2(I;L6(Ω;R3))

‖v‖1/4L∞(I;L2(Ω;R3))

‖∇z‖L4(I;L2(Ω;R3×3))‖u‖3/4L2(I;L6(Ω;R3))‖u‖1/4

L∞(I;L2(Ω;R3)).

14We use Proposition 1.41 for p1 = q2 = 2, q1 = 6, p2 = +∞, and λ = 1/2; cf. also Exam-ple 8.77.


Let us now abbreviate W1 := W 1,2,4/3(I;W 1,20,div(Ω;R

n),W 1,20,div(Ω;R

n)∗) ∩L∞(I;L2(Ω;Rn)) and W2 := W 1,2,4/3(I;W 1,2(Ω),W 1,2(Ω)∗) ∩ L∞(I;L2(Ω)). Wedefine the mapping

M : W1×W2 ⇒ W1×W2 (12.55)

as M(v, ϑ) being the set of very weak solutions (u, θ) to (12.47)–(12.48) satisfyingthe bounds (12.49).

Lemma 12.15 (Continuity). Let n ≤ 3 and (12.46) hold. The set-valued mappingM : W1×W2 ⇒ W1×W2, see (12.55), is weakly upper semi-continuous.

Proof. Assume (vk, θk)∗⇀ (v, θ) in W1 × W2. By the “interpolated” Aubin-Lions’

Lemma 7.8, we have vk → v in L2�−ε(Q;Rn), cf. (8.152); if n ≤ 3, we can consider2�≥10/3, cf. (8.131). As 2� > 2, we certainly have (vk · ∇)uk ⇀ (v · ∇)u weaklyin L1(Q;Rn). Similarly, we have also vk · ∇θk ⇀ v · ∇θ weakly in L1(Q). Hencewe can make the limit passage just by weak continuity. �

Proposition 12.16 (Existence of a fixed point). Let n ≤ 3 and (12.46) hold.The set-valued mapping M has a fixed point (u, θ) ∈ M(u, θ) which is a very weaksolution to (12.44)–(12.45).

Proof. The closed bounded convex set{(u, θ) ∈ W1 × W2;

∥∥u∥∥L2(I;W 1,2

0,div(Ω;Rn))∩L∞(I;L2(Ω;Rn))≤ C1(1+C3),∥∥∥∂u

∂t

∥∥∥L4/3(I;W 1,2

0,div(Ω;Rn)∗)≤ C2(1+C1+C3+C1C3),∥∥θ∥∥

L2(I;W 1,2(Ω))∩L∞(I;L2(Ω))≤ C3,∥∥∥∂θ

∂t

∥∥∥L4/3(I;W−1,2(Ω))

≤ C4(1+C1+C1C3)}

(12.56)

is weakly* compact in W1 × W2. Due to Lemma 12.14, M maps this set intoitself. As (12.47)–(12.48) are linear and (12.49) are convex inequalities, the valuesof M are convex. By Lemma 12.14, the values of M are nonempty. Taking intoaccount Lemma 12.15, we can use Kakutani fixed-point theorem 1.11 to get (u, θ) ∈M(u, θ). �

Exercise 12.17. Apply Galerkin’s method to (12.44) and prove convergence of theapproximate solutions and thus existence of the very weak solution to (12.44)without using Schauder’s-type fixed-point theorem.

12.3 Predator-prey system

Let us deal with a special evolution variant of the Lotka-Volterra system (6.51):15

15This special model and its analysis is after [346].

12.3. Predator-prey system 409

∂u

∂t− d1Δu = u

(a1 − b1u− c1v

)∂v

∂t− d2Δv = v

(a2 − c2u

)⎫⎪⎬⎪⎭ in Q,

u(t, ·)|Γ = 0, v(t, ·)|Γ = 0 on Σ,

u(0, ·) = u0, v(0, ·) = v0 on Ω.

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭(12.57)

We consider the constants corresponding to the predator-prey variant, i.e. b1 ≥ 0,d1, d2 > 0, a1, c1 > 0 and a2, c2 < 0, so that u and v represent a prey and apredator densities, respectively; then a1 is the growth rate of the prey speciesin the absence of the predators while −a2 is the death rate of the predators inthe absence of the prey; for further interpretation see Section 6.3. The maximalconcentration of preys in the absence of predators (i.e. the carrying capacity of theenvironment) is a1/b1 =: γ1. Instead of using Schauder’s fixed-point theorem asin Section 6.3, we can now be more constructive and use the semi-implicit Rothemethod: to be more specific, we seek uk

τ , vkτ ∈ W 1,2

0 (Ω) satisfying

ukτ − uk−1

τ

τ− d1Δuk

τ = uk−1τ a1

(1− b1u

kτ

)− c1ukτv

k−1τ , (12.58a)

vkτ − vk−1τ

τ− d2Δvkτ = vkτ

(a2 − c2u

kτ

)(12.58b)

for k = 1, . . . , T/τ , while for k = 0 we consider

u0 = u0τ , v0 = v0τ , (12.59)

with some u0τ , v0τ ∈ W 1,20 (Ω) such that u0τ → u0, v0τ → v0 in L2(Ω) and

‖u0τ‖W 1,20 (Ω) = O(1/

√τ ), and ‖v0τ‖W 1,2

0 (Ω) = O(1/√τ). Note that the boundary-

value problems (12.58) are linear and (12.58b) is decoupled from (12.58a), whichsuggests an efficient numerical strategy after a further discretization by a Galerkinmethod.

Lemma 12.18 (A-priori bounds). If τ ≤ τ0 < 1/(2a2−2c2γ1)+ with γ1 = a1/b1,

the elliptic problems in (12.58) have unique solutions (ukτ , v

kτ ) ∈ W 1,2

0 (Ω)2 for allk = 1, . . . , T/τ , which satisfy 0 ≤ uk

τ ≤ γ1 and 0 ≤ vkτ provided 0 ≤ u0 ≤ γ1 and0 ≤ v0 ∈ L2(Ω). Furthermore, the following a-priori estimates hold:

∥∥(uτ , vτ )∥∥L2(I;W 1,2

0 (Ω))2≤ C,

∥∥∥(∂uτ

∂t,∂vτ∂t

)∥∥∥L2(I;W−1,2(Ω))2

≤ C. (12.60)

Proof. We use an induction argument for the first part of the lemma. Let ussuppose that uk−1

τ , vk−1τ ∈L∞(Ω) satisfy 0 ≤ uk−1

τ ≤ γ1 and 0 ≤ vk−1τ . Then, we

have to prove that ukτ and vkτ belong to W 1,2

0 (Ω) and inherit these bounds.The linear problem (12.58a) for uk

τ is coercive on W 1,20 (Ω) because

b1, c1, uk−1τ , vk−1

τ ≥ 0, so that by Lax-Milgram’s Theorem 2.19 it possesses a unique


weak solution. Let us show that ukτ ≥ 0. Testing the weak formulation of (12.58a)

by (ukτ )−, one gets

1

τ

∥∥(ukτ )−∥∥2

L2(Ω)+ d1

∥∥∇(ukτ )−∥∥2

L2(Ω;Rn)≤∫Ω

(1τ+ a1

)uk−1τ (uk

τ )− dx ≤ 0. (12.61)

Hence, we get (ukτ )− = 0. Let us further prove that uk

τ ≤ γ1. Testing the weakformulation of (12.58a) by (uk

τ − γ1)+, we obtain

∥∥(ukτ−γ1)

+∥∥2L2(Ω)

+ τd1∥∥∇(uk

τ−γ1)+∥∥2L2(Ω;Rn)

=

∫Ω

(uk−1τ −γ1)(u

kτ−γ1)

+

−τ(a1y

k−1τ

ukτ−γ1γ1

(ukτ−γ1)

+ + c1ukτ (u

kτ−γ1)

+vk−1τ

)dx ≤ 0 (12.62)

since 0 ≤ uk−1τ ≤ γ1 and vk−1

τ ≥ 0. Hence, ukτ ≤ γ1 a.e. in Ω.

Now, (12.58b) is a linear boundary-value problem for vkτ which is coerciveon W 1,2

0 (Ω) if the coefficient 1τ − a2 + c2u

kτ is non-negative. Taking into account

uk ≤ γ1, it needs the condition τ < 1/(a2 − c2γ1)+. Therefore, by Lax-Milgram’s

Theorem 2.19, it possesses a unique solution vkτ ∈ W 1,20 (Ω). Let us show that

vkτ ≥ 0. Testing the weak formulation of (12.58b) by (vkτ )−, one gets

(1τ−a2+c2γ1

)∥∥(vkτ )−∥∥2L2(Ω)+ d2

∥∥∇(vkτ )−∥∥2

L2(Ω;Rn)≤∫Ω

vk−1τ

τ(vkτ )

− dx ≤ 0;

recall that c2 ≤ 0. Hence, if τ < 1/(a2 − c2γ1)+, we get (vkτ )

− = 0.

Let us now prove the estimates (12.60). Testing (12.58a) by ukτ and using

Young’s inequality and non-negativity of vk−1τ , we have

1

2τ‖uk

τ‖2L2(Ω) + d1‖∇ukτ‖2L2(Ω;Rn) ≤

∫Ω

(1τ+

a12

)(uk−1

τ )2 +a12(uk

τ )2dx. (12.63)

Summing it for k = 1, . . . , T/τ yields boundedness of uτ and, by using the tech-nique of combination (8.18) with (8.38), also of uτ in L2(I;H1

0 (Ω)).

Now, testing (12.58b) by vkτ and using Young inequality we also have

1

τ

∥∥vkτ∥∥2L2(Ω)+ d2

∥∥∇vkτ∥∥2L2(Ω;Rn)

≤∫Ω

1

τvk−1τ vkτ + (a2 − c2u

kτ )(v

kτ )

2 dx

≤ 1

2τ

∥∥vk−1τ

∥∥2L2(Ω)

+1

2τ

∥∥vkτ∥∥2L2(Ω)+ (a2 − c2γ1)

∥∥vkτ∥∥2L2(Ω). (12.64)

By using the discrete Gronwall inequality (1.70), we get vτ and, by using again thetechnique of combination (8.18) with (8.38), also vτ bounded in L2(I;W 1,2

0 (Ω))independently of τ provided τ ≤ τ0 < 1/(2a2 − 2c2γ1)

+, as assumed.

12.3. Predator-prey system 411

Using the “retarded” function uRτ as defined in (8.202) and analogously for

vRτ , the scheme (12.58) can be written down in a “compact” form as

∂uτ

∂t− d1Δuτ = a1u

R

τ

(1− uτ

γ1

)− c1uτ v

R

τ , (12.65a)

∂vτ∂t

− d2Δvτ = a2vτ − c2uτ vτ . (12.65b)

In view of this, we can estimate∥∥∥∂uτ

∂t

∥∥∥2L2(I;W−1,2(Ω))

= sup‖z‖

L2(I;W1,20

(Ω))≤1

∫Q

d1∇uτ ·∇z + a1uR

τ

(1− uτ

γ1

)z

−c1uτ vR

τ z dxdt ≤ d1∥∥∇uτ

∥∥L2(Q;Rn)

+measn+1(Q)1/2a1γ14

+ c1γ1∥∥vR

τ

∥∥L2(Q)

which bounds ∂∂tuτ . Similarly, from (12.65), one obtains∥∥∥∂vτ

∂t

∥∥∥2L2(I;W−1,2(Ω))

= sup‖z‖

L2(I;W1,20 (Ω))

≤1

∫Q

d1∇vτ ·∇z + a2vτz

−c2uτ vτz dxdt ≤ d1∥∥∇vτ

∥∥L2(Q;Rn)

+max(|a2 − c2γ1|,−a2)∥∥vτ∥∥L2(Q)

. �

Proposition 12.19 (Convergence, uniqueness). For τ ↘ 0, (uτ , vτ ) ⇀ (u, v)weakly in W 2 with W := W 1,2,2(I;W 1,2

0 (Ω),W−1,2(Ω)) and (u, v) is the uniqueweak solution to (12.57).

Proof. The mentioned converging (sub)sequence does exist thanks to (12.60). ByAubin-Lions’ lemma, also (uτ , vτ ) → (u, v) in L2(Q)2. By interpolation and by(8.50), (8.31), and (12.60), it holds that∥∥uτ−uτ

∥∥L2(Q)

≤ C0

∥∥uτ−uτ

∥∥1/2L2(I;W 1,2

0 (Ω))

∥∥uτ−uτ

∥∥1/2L2(I;W−1,2(Ω))

≤ C0

√2∥∥uτ

∥∥1/2L2(I;W 1,2

0 (Ω))

√τ

3

∥∥∥∂uτ

∂t

∥∥∥1/2L2(I;W−1,2(Ω))

= O(√τ). (12.66)

Hence uτ → u strongly in L2(Q). By analogous arguments, also vτ → v in L2(Q).

Now we are to make a limit passage in (12.65) just by weak continuity.Note that the nonlinearity R2

+ × R2 → R2 : (r1, r′1, r2, r

′2) �→

(a1r

′1(1 − r1/γ1) −

c1r1r′2, a2r2 − c2r1r2

)has at most a quadratic growth so that, by continuity of

the respective Nemytskiı mapping L2(Q)4 → L1(Q)2, the right-hand sides of(12.65a,b) converge strongly in L1(Q) to a1u(1 − u/γ1) − c1uv and a2v − c2uv,respectively. The limit passage in the left-hand-side terms in (12.65a,b) is obviousbecause they are linear. Note that the mapping u �→ u(0, ·) : W → L2(Ω) is weaklycontinuous which allows us to pass to the limit in the respective initial conditions.


To prove uniqueness of the solution to (12.57), we consider two weak solutions(u1, v1) and (u2, v2), subtract the corresponding equations and test them by u12 :=u1 − u2 and v12 := v1 − v2, respectively. This gives

1

2

d

dt

(∥∥u12

∥∥2L2(Ω)

+∥∥v12∥∥2L2(Ω)

)+ d1

∥∥∇u12

∥∥2L2(Ω;Rn)

+ d2∥∥∇(v12)

∥∥2L2(Ω;Rn)

+

∫Ω

b1(u1 + u2

)(u12)

2dx− a2∥∥v12∥∥2L2(Ω)

= a1∥∥u12

∥∥2L2(Ω)

−∫Ω

c1(u1v1−u2v2

)u12 − c2

(u1v1−u2v2

)v12dx

≤ a1∥∥u12

∥∥2L2(Ω)

+ c1∥∥u1

∥∥L∞(Ω)

∥∥v12∥∥L2(Ω)

∥∥u12

∥∥L2(Ω)

+c2∥∥u1

∥∥L∞(Ω)

∥∥v12∥∥2L2(Ω)+ c2

∥∥u12

∥∥L2(Ω)

∥∥v2∥∥L∞(Ω)

∥∥v12∥∥L2(Ω),

where we also used that u1, u2, v1, and v2 are non-negative and that u1 and v2have upper bounds. Then, by Young’s and Gronwall’s inequalities, we get u12 = 0and v12 = 0. Thus we showed the uniqueness and thus the convergence of thewhole sequence {(uτ , vτ )}τ>0. �

Exercise 12.20. Having (12.60) at disposal, execute the test of (12.58) by ukτ−uk−1

τ

and vkτ−vk−1τ to prove the a-priori bound of uτ and vτ in W 1,2(I;L2(Ω)) ∩

L∞(I;W 1,p0 (Ω)).16

12.4 Semiconductors

Modelling of transient regimes of semiconductor devices conventionally relies onthe evolution variant of Roosbroeck’s drift-diffusion system (6.68), i.e.

div(ε∇φ

)= n− p+ cD in Q, (12.67a)

∂n

∂t− div

(∇n− n∇φ)= r(n, p) in Q, (12.67b)

∂p

∂t− div

(∇p+ p∇φ)= r(n, p) in Q, (12.67c)

where we use the conventional notation of Section 6.5 except the sign conventionof r. We can see that the magnetic field is still neglected and the electric fieldφ, which varies much faster than the carrier concentrations n and p, is governed

16Hint: In terms of the interpolants, test (12.65) respectively by ∂∂t

uτ and ∂∂t

vτ to obtain∫ t0 ‖ ∂

∂tuτ‖2L2(Ω)

+‖ ∂∂t

vτ ‖2L2(Ω)dt+ 1

2d1‖∇uτ (t)‖2L2(Ω;Rn)

+ 12d2‖∇vτ (t)‖2L2(Ω;Rn)

≤ ∫ t0 f1

∂∂t

uτ+

f2∂∂t

vτ dt+ 12d1‖∇u0‖2L2(Ω;Rn)

+ 12d2‖∇v0‖2L2(Ω;Rn)

for t = kτ , k = 1, ..., T/τ , where f1 and f2

denote the right-hand sides of (12.65a) and (12.65b), respectively. Realize that, by (12.60), we

have already estimated f1, f2 ∈ L∞(I;L2∗ (Ω)).


by the quasistatic equation (12.67a) which therefore does not involve any timederivative of φ.

Of course, (12.67b,c) is to be completed by initial conditions

n(0, ·) = n0, p(0, ·) = p0, (12.68)

and some boundary conditions; e.g. Dirichlet ones of ΓD with measn−1(ΓD) > 0(electrodes with time-varying voltage) and zero Neumann one on ΓN = Γ \ΓD (anisolated part), i.e.

φ|ΣD = φΣ|ΣD , n|ΣD = nΣ|ΣD , p|ΣD = pΣ|ΣD on ΣD := (0, T )×ΓD, (12.69a)

∂φ

∂ν=

∂n

∂ν=

∂p

∂ν= 0 on ΣN := (0, T )×ΓN, (12.69b)

with nΣ and pΣ constant in time, i.e. nΣ(t, ·) = nΓ and pΣ(t, ·) = pΓ. Again, wemade an exponential-type transformation but now slightly different than (6.70)17,namely we introduce a new variable set (φ, u, v) related to (φ, n, p) by

n = eu , p = ev (12.70)

and abbreviates(u, v) := r

(eu, ev

). (12.71)

Obviously, (12.70) transforms the currents jn = ∇n − n∇φ = eu∇(u − φ) andjp = −∇p − p∇φ = −ev∇(v + φ). Another elegant trick18, proposed by Gajew-ski [164, 165], consists in time-differentiation of (12.67a), which leads, by using(12.67b,c) together with the fact that concentration of dopants cD = cD(x) is time-independent, to the pseudoparabolic equation ∂

∂t (−div(ε∇φ)) = ∂∂t (p−n− cD) =

div(jn−jp). Of course, now we need the initial condition for φ, namely φ(0, ·) = φ0,with φ0 satisfying

div(ε∇φ0

)= n0 − p0 + cD on Ω; (12.72)

and the initial conditions (12.68) now transform to

u0 = ln(n0) , v0 = ln(p0). (12.73)

Hence the system (12.67) transforms to

∂

∂t

(div(ε∇φ)

)+ div

(eu∇(φ− u) + ev∇(φ + v)

)= 0 (12.74a)

∂

∂t

(eu)− div

(eu∇(u− φ)

)= s(u, v), (12.74b)

∂

∂t

(ev)− div

(ev∇(v + φ)

)= s(u, v), (12.74c)

17Realize that (6.70) would not result in a doubly-nonlinear structure like (11.65).18For alternative analysis of (12.67) without differentiating (12.67a) in time see e.g. [166, 167]

or [119] or [276, Sect.3.7].


while the boundary conditions (12.69) transform to

φ(t, ·)|ΣD = φΣ(t, ·)|ΣD , u(t, ·)|ΓD = uΓ|ΓD , v(t, ·)|ΓD = vΓ|ΓD on ΓD,

(12.75a)

∂φ

∂ν=

∂u

∂ν=

∂v

∂ν= 0 on ΣN, (12.75b)

where uΓ := ln(nΓ) and vΓ := ln(pΓ).The weak solution to (12.67) with (12.68)-(12.69) is understood as (φ, u, v) ∈

L2(I;W 1,2(Ω))3 such that also ∂∂t (div(ε∇φ)), ∂

∂teu, ∂

∂tev ∈ L2(I;W 1,2

0 (Ω)∗),(12.75a) holds, u(0, ·) = u0 and v(0, ·) = v0 with u0 and v0 from (12.73), andthe integral identity⟨ ∂

∂t

(div(ε∇φ)

), z1

⟩+⟨ ∂

∂t

(eu), z2

⟩+⟨ ∂

∂t

(ev), z3

⟩=

∫Ω

eu∇(u−φ)·∇(z1−z2)− ev∇(φ+v)·∇(z1+z3) + s(u, v)(z2+z3) dx

(12.76)

holds for a.a. t ∈ I and all z ∈ W 1,2(Ω)3 such that z|ΓD = 0.The analysis of the original model is complicated and we confine ourselves

only to a modified model arising by truncation of the nonlinearity ξ �→ eξ andthen also of s(·, ·), i.e. by replacing them by

el(ξ) := emin(l,max(−l,ξ)) , sl(u, v) := r(el(u), el(v)

); (12.77)

here l is a positive constant. Hence el(r) = er for r ∈ [−l, l]. We will analyze itby the Galerkin approximation by using the subspaces Vk of W 1,2

ΓD(Ω) := {v ∈

W 1,2(Ω); v|ΓD = 0}, and assuming, for simplicity, that u0, v0, uΓ, vΓ, φΣ(t, ·) ∈ Vk

for any k, while φ0 has to be approximated by a suitable φ0k ∈ Vk. Note that elmodifies the original nonlinearity out of the interval [−l, l] and makes, in particular,sl bounded. Hence we get an approximate solution (φkl, ukl, vkl).

Lemma 12.21 (A-priori bounds). Let uΓ, vΓ ∈ W 1,2(Ω), φΣ ∈ L∞(I;W 1,2(Ω)).Then, for l ∈ N fixed, the approximate solution (φkl, ukl, vkl) satisfies:∥∥φkl

∥∥L∞(I;W 1,2(Ω))

≤ Cl, (12.78a)∥∥ukl

∥∥L2(I;W 1,2(Ω))

≤ Cl,∥∥vkl∥∥L2(I;W 1,2(Ω))

≤ Cl, (12.78b)∥∥el(ukl)∥∥L∞(Q)

≤ Cl,∥∥el(vkl)∥∥L∞(Q)

≤ Cl, (12.78c)∥∥∥ ∂

∂t

(div(ε∇φ)

)∥∥∥L2(I;W−1,2(Ω))

≤ Cl, (12.78d)∥∥∥ ∂

∂tel(ukl)

∥∥∥L2(I;W−1,2(Ω))

≤ Cl,∥∥∥ ∂

∂tel(vkl)

∥∥∥L2(I;W−1,2(Ω))

≤ Cl. (12.78e)


Proof. Let us test (12.74) modified as outlined above by ([φkl−φΣ](t, ·), ukl(t, ·)−uΓ, vkl(t, ·)−vΓ) itself. This, after integration over [0, t] and the by-part integration∫ t

0∂∂tel(ukl)uΓdt = (el(ukl(t, ·)) − el(u0))uΓ, gives∫Ω

[ ε2|∇φkl|2+ [el]

∗(el(ukl))+ [el]

∗(el(vkl))](t, ·) dx+

∫ t

0

∫Ω

el(ukl)∣∣∇(ukl−φkl)

∣∣2 + el(vkl)∣∣∇(vkl+φkl)

∣∣2 dxdt=

∫Ω

(ε2|∇φ0|2+ [el]

∗(el(u0))+ [el]

∗(el(v0))+ ε[∇φkl·∇φΣ](t, ·)

− ε∇φ0k·∇φΣ(0, ·) +(el(ukl(t, ·))− el(u0)

)uΓ +

(el(vkl(t, ·))− el(v0)

)vΓ

)dx

+

∫ t

0

∫Ω

sl(ukl, vkl)(ukl + vkl)− ε∇φkl ·∇∂φΣ

∂t

+ el(ukl)∇(φkl−ukl) ·∇(φΣ−uΓ) + el(vkl)∇(φkl+vkl) ·∇(φΣ+vΓ) dxdt

where [el]∗(·) is the Legendre-Fenchel conjugate to the primitive function el : ξ �→∫ ξ

0 el(ζ)dζ to el, cf. the formula (11.70). We further use the estimates∫Ω

el(ukl)∣∣∇(ukl − φkl)

∣∣2dx ≥∫Ω

el(ukl)(12

∣∣∇ukl

∣∣2 − ∣∣∇φkl

∣∣2)dx≥∫Ω

e−l

2

∣∣∇ukl

∣∣2 − el∣∣∇φkl

∣∣2dx, (12.79)

which yields the term el|∇φkl|2 to be treated “on the right-hand side” by Gron-wall’s inequality. Similarly,∫

Ω

el(vkl)∣∣∇(vkl + φkl)

∣∣2dx ≥∫Ω

e−l

2

∣∣∇vkl∣∣2 − el

∣∣∇φkl

∣∣2dx. (12.80)

Using the obvious estimate el(ξ) ≤ el|ξ| and thus [el]∗(ζ) ≥ δ[−el,el], by Gronwall’s

inequality we eventually obtain the estimates (12.78a-c). Furthermore, from theequations themselves we obtain dual estimates on the time derivatives (12.78d,e).

�

Proposition 12.22 (Convergence). The approximate Galerkin solution(φkl, ukl, vkl) converges (as a subsequence) for k → ∞ (l kept fixed) weakly inL2(I;W 1,2(Ω))3 to a weak solution, let us denote it by (φl, ul, vl), of (12.74) witheu and ev replaced by el(u) and el(v), respectively.

Proof. The declared convergence can be shown in parallel to Section 11.2.1. Inparticular, by Aubin-Lions’ lemma we have wkl := el(ukl) → wl in L2(Q); herewe also use that ∇wkl = e′l(ukl)∇ukl is bounded in L2(Q;Rn) due to (12.78b).Also ukl ⇀ ul in L2(Q), thus

∫Q wklukl dxdt →

∫Q wlul dxdt so that by maximal


monotonicity of el one can conclude that wl := el(ul). Similarly, we can also proveel(vkl) → el(vl) in L2(Q), and therefore also sl(ukl, vkl) = r(el(ukl), el(vkl)) →r(el(ul), el(vl)) = sl(ul, vl). Then one can pass to the limit in the Galerkin identity

limk→∞

∫Q

∂wkl

∂tz + el(ukl)∇(ukl − φkl

) · ∇z − sl(ukl, vkl)z dxdt

=

∫ T

0

(⟨∂wl

∂t, z⟩+

∫Ω

el(ul)∇(ul − φl

) · ∇z − sl(ul, vl)z dx

)dt. (12.81)

Analogous limit passage can be made in the other equations; the term∂∂t (div(ε∇φkl)) is linear hence the limit passage is possible by a weak convergencedue to the estimate (12.78d). �

Remark 12.23 (Limit passage for l → +∞). The strategy to pass to the originalsystem (12.74) is to show a-priori bounds for ul and vl in L∞(Q) independent ofl and then, if l is chosen bigger than these bounds, (φl, ul, vl) is the weak solutionof the non-modified system (12.74). For this, rather nontrivial step, we refer toGajewski [163] and Gajewski and Groger [167].

Remark 12.24 (Index-2 differential-algebraic system). When applying Galerkinapproximation to (12.67), we obtain a system of so-called differential-algebraicequations (DAEs), i.e. time derivative is involved only in some components (herecorresponding to u and v), while the rest (here corresponding to φ-components)forms an algebraic system. If one can eliminate the algebraic part after differentiat-ing it k-times, we say that the DAEs have the (differential) index k+1. Therefore,as Gajewski’s transformation (12.74) shows, the original system (12.67) can beviewed (in its Galerkin approximation) as an index-2 DAE.

Remark 12.25 (Nernst-Planck-Poisson system). The special case cD ≡ 0 and r ≡ 0is a basic model for electro-diffusion of ions in electrolytes, which was formulatedby W. Nernst and M. Planck at the end of the 19th century.19

12.5 Phase-field model

To describe solidification/melting processes at the microscale, models of a so-called phase-field type can be used. A basic Caginalp’s model [87]20 consists of the

19W.H. Nernst received the Nobel prize in chemistry for his work in thermochemistry in 1920(while also M.K.E.L. Planck received Nobel’s prize already in 1918 in physics but not directlyrelated to the system (12.67) with cD = r = 0).

20For further study see Brokate and Sprekels [71, Sect.6.2], Elliott and Zheng [137], Kenmochiand Niezgodka [229], Zheng [428, Sect.4.1].

12.5. Phase-field model 417

following system:

∂θ

∂t= Δθ +

∂v

∂t+ g,

∂v

∂t= Δv − c(v)− θ

⎫⎪⎬⎪⎭ in Q,

θ = 0, v = 0 on Σ,

θ(0, ·) = θ0, v(0, ·) = v0 on Ω

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭(12.82)

where θ plays the role of a temperature and the “order parameter” v distin-guishes particular phases according to where the (typically nonconvex) potentialc of c : R → R attains its minima. This is an interesting system not only for itsapplications but also for “training” purposes because there are various ways toget a-priori estimates and then prove existence of a solution. Let us outline thea-priori estimates heuristically.

First option: Summing the equations in (12.82) gives ∂∂tθ = Δ(θ+v)−c(v)−

θ + g, and then testing it by θ yields the estimate

1

2

d

dt

∥∥θ∥∥2L2(Ω)

+∥∥∇θ

∥∥2L2(Ω;Rn)

+∥∥θ∥∥2

L2(Ω)=

∫Ω

∇θ·∇v +(g − c(v)

)θ dx

≤ 1

2

∥∥∇θ∥∥2L2(Ω;Rn)

+1

2

∥∥∇v∥∥2L2(Ω;Rn)

+3

2

∥∥θ∥∥2L2(Ω)

+∥∥g∥∥2

L2(Ω)+∥∥c(v)∥∥2

L2(Ω).

(12.83)

Then, assuming c(v)v ≥ −c1− c2v2, c1, c2 ≥ 0, and testing the second equation in

(12.82) by v yields:

1

2

d

dt

∥∥v∥∥2L2(Ω)

+∥∥∇v

∥∥2L2(Ω;Rn)

=

∫Ω

(θ − c(v)

)v dx

≤ c1measn(Ω) +1

2

∥∥θ∥∥2L2(Ω)

+(12+ c2

)∥∥v∥∥2L2(Ω)

. (12.84)

Summing (12.83) and (12.84), by standard procedure via Gronwall’s inequalityone obtains the a-priori estimates∥∥θ∥∥

L∞(I;L2(Ω))≤ C,

∥∥∇θ∥∥L2(Q;Rn)

≤ C, (12.85)∥∥v∥∥L∞(I;L2(Ω))

≤ C,∥∥∇v

∥∥L2(Q;Rn)

≤ C, (12.86)

provided θ0, v0 ∈L2(Ω), g∈L2(Q), and c(·) has at most linear growth because ofthe last term in (12.83).21 The “dual” estimates of ∂

∂tθ and ∂∂tv in L2(I;W 1,2(Ω)∗)

then follow standardly.

21Standardly, c is of the type c(r) = (r2 − 1)2, which is not consistent with this approach,however.


Second option: Testing the first equation of (12.82) by θ and the second oneby ∂

∂tv gives

1

2

d

dt

∥∥θ∥∥2L2(Ω)

+∥∥∇θ

∥∥2L2(Ω;Rn)

=

∫Ω

∂v

∂tθ + gθ dx ≤

∫Ω

∂v

∂tθ dx

+ ε∥∥θ∥∥2

L2∗ (Ω)+ Cε

∥∥g∥∥2L2∗′(Ω)

, (12.87)∥∥∥∂v∂t

∥∥∥2L2(Ω)

+1

2

d

dt

∥∥∇v∥∥2L2(Ω)

+d

dt

∫Ω

c(v) dx = −∫Ω

θ∂v

∂tdx , (12.88)

where c : R → R is the potential (i.e. the primitive function) of c. Supposingθ0∈L2(Ω), v0∈W 1,2(Ω), g∈L2(I;L2∗′

(Ω)), and c ≥ 0, and summing (12.87) and(12.88), and using Gronwall’s inequality eventually yields the estimates:∥∥θ∥∥

L∞(I;L2(Ω))≤ C,

∥∥∇θ∥∥L2(Q;Rn)

≤ C, (12.89)∥∥∇v∥∥L∞(I;L2(Ω;Rn))

≤ C,∥∥∥∂v∂t

∥∥∥L2(Q)

≤ C. (12.90)

Coming back to the first equation of (12.82), one gets standardly the “dual” esti-mate of ∂

∂tθ in L2(I;W 1,2(Ω)∗).Third option: Testing the second equation in (12.82) again by ∂

∂tv but the

first one by ∂∂tθ gives, besides (12.88), the estimate∥∥∥∂θ

∂t

∥∥∥2L2(Ω)

+1

2

d

dt

∥∥∇θ∥∥2L2(Ω;Rn)

=

∫Ω

(∂v∂t

+ g)∂θ∂t

dx

≤ 1√2

(∥∥∥∂θ∂t

∥∥∥2L2(Ω)

+∥∥∥∂v∂t

∥∥∥2L2(Ω)

+∥∥g∥∥2

L2(Ω)

). (12.91)

Supposing θ0, v0∈W 1,2(Ω), and g∈L2(Q) and summing (12.88) and (12.91), andestimating | ∫Ω θ ∂

∂tv dx| ≤ N2‖∇θ‖2L2(Ω;Rn) +14‖ ∂

∂tv‖2L2(Ω) in (12.88) with N the

norm of the embedding W 1,2(Ω) ⊂ L2(Ω) gives via Gronwall’s inequality againthe estimates (12.90) together with∥∥θ∥∥

L∞(I;W 1,2(Ω))≤ C,

∥∥∥∂θ∂t

∥∥∥L2(Q)

≤ C. (12.92)

Exercise 12.26 (Scaling). Modify the above estimate for the scaling by ζ and ξ asin (9.80) similarly as done in Example 9.32.

Exercise 12.27 (Galerkin’s method). Prove existence of a weak solution to (12.82)by convergence of the approximate solution constructed by the Galerkin methodbased on the estimates (12.83)–(12.84), or (12.87)–(12.88), or (12.88)–(12.91). Notethat Aubin-Lions’ Lemma 7.7 is used for the only nonlinear term c.22

22Hint in case of (12.87)–(12.88): Make a growth assumption |c(r)| ≤ C(1 + |r|2∗−ε), ε > 0,so that, in view of (12.90) and Aubin-Lions’ Lemma 7.7, Galerkin’s sequence of v’s is compact

in L2∗−ε(Q) and the limit passage through the nonlinear term c is possible as the Nemytskiı

mapping Nc : L2∗−ε(Q)→ L1(Q) is continuous.

12.5. Phase-field model 419

Exercise 12.28 (Semi-implicit Rothe method). An advantageous modification ofRothe’s method that would lead to a de-coupling of the system at each time levelcan be based on the semi-implicit formula:

θkτ−θk−1τ

τ= Δθkτ +

vkτ−vk−1τ

τ+ gkτ ,

vkτ−vk−1τ

τ= Δvkτ − c(vkτ ) + θk−1

τ . (12.93)

Modify the a-priori estimates (12.88) and (12.91) and prove the convergence.23

Exercise 12.29 (Penrose and Fife’s generalization [333]). Consider

∂θ

∂t= Δβ(θ) + d(v)

∂v

∂t+ g,

∂v

∂t= Δv − c(v)− d(v)β(θ)

⎫⎪⎬⎪⎭ (12.94)

with β : R → R increasing. Note that for d(·) = 1 and β(r) = r, we get (12.82).The physically justified option suggested by Penrose and Fife [333] uses β(r) =−1/r, which requires quite sophisticated techniques,24 however. Introducing thenew variable u = β(θ) and e(·) = β(·)−1, (12.94) transforms to a doubly-nonlinearsystem ∂

∂te(u) −Δu = d(v) ∂∂tv + g and ∂

∂tv − Δv = −c(v) − d(v)u. Assume, forsimplicity, inf e′(·) ≥ ε > 0 and qualify also c(·) and d(·) appropriately, and use thetechnique from Sect. 11.2.1 to get the a-priori estimate of u in L2(I;W 1,2(Ω)) ∩L∞(I;L2(Ω)) and of v in W 1,∞,2(I;W 1,2(Ω), L2(Ω)) by testing these equationsby u and ∂

∂tv, respectively.25 After deriving still a dual estimate for ∂

∂te(u), proveconvergence of, say, Rothe approximations as in Sect. 11.2.1.

23Hint: Modification of the a-priori estimates (12.88) and (12.91) can be based sim-

ply on θk−1τ = θkτ − τ

θkτ−θk−1τ

τso that one can estimate ‖θk−1

τ ‖2L2(Ω)

≤ 2‖θkτ ‖2L2(Ω)+

2τ2‖ θkτ−θk−1τ

τ‖2L2(Ω)

. In other words, ‖θRτ ‖2L2(Ω)

≤ 2‖θτ‖2L2(Ω)+2τ2‖ ∂

∂tθτ‖2L2(Ω)

where the “re-

tarded” Rothe function θRτ is as in (8.202). The additional term τ2‖ ∂

∂tθτ‖2L2(Ω)

can be absorbed

if τ is small enough. The convergence of the scheme based on (12.93), i.e.

∂θτ

∂t= Δθτ +

∂vτ

∂t+ gτ ,

∂vτ

∂t= Δvτ − c(vτ ) + θR

τ ,

can be proved as in Exercise 12.27 when using also ‖θRτ −θτ‖L2(Ω) = τ‖ ∂

∂tθτ‖L2(Ω) = O(τ).

24See Brokate and Sprekels [71, Sect.6.3], Colli and Sprekels [103], Elliott and Zheng [137], orZheng [428, Sect.4.1.2]. Asymptotic behaviour under scaling of the terms ∂

∂tv and Δv in (12.94)

and relation to a modified Stefan problem is in [103].25Hint: Use

∫Ω( ∂∂t

e(u))u dx = ddt

∫Ω[ e ]∗(u) dt with [ e ]∗ the conjugate function of the primitive

function of e satisfying [ e ]∗(r) ≤ |r|2/(2ε) because e (r) ≥ ε|r|2/2, cf. (8.248)–(8.249), and notethat the arisen terms ± ∫

Ω u d(v) ∂∂t

v dx cancel with each other, and eventually estimate

d

dt

([ e ]∗(u) +

1

2|∇v|2 + c (v)

)+ |∇u|2 +

∣∣∣∂v∂t

∣∣∣2 =

∫Ωgudx ≤ ∥∥g∥∥

L2∗′(Ω)

∥∥u∥∥L2∗ (Ω)

.


Exercise 12.30 (Kenmochi-Niezgodka’s modification [228, 230]26). Consider aCahn-Hilliard equation ∂

∂tv+Δ(Δv− c(v)− θ) = 0 instead of ∂∂tv = Δv− c(v)− θ

in (12.82) and derive the a-priori estimates by testing by θ and by Δ−1 ∂∂tv.

27

Exercise 12.31 (Benes’ generalizations [43, 44]). Augment the second equationin (12.82) to ∂

∂tv = Δv − c(v) − θ + ψ(θ)|∇v| with ψ : R → R continuous andbounded, and modify all above estimates28 and prove convergence of the Galerkinapproximation29. Consider further a given velocity field �v : Q → Rn and augment(12.82) by the advection terms, i.e. ∂

∂tθ + �v · ∇θ and ∂∂tv + �v · ∇v instead of ∂

∂tθ

and ∂∂tv, respectively, and modify all above estimates30 and prove convergence of

the Galerkin approximation31.

12.6 Navier-Stokes-Nernst-Planck-Poisson-type system

An incompressible ionized mixture of L mutually reacting chemical constituentsoccurs in various biological or chemical applications. We accept a so-called Eckart–

26For such a sort of model see also Alt and Paw�low [12].27Hint: Test the first equation in (12.82) by θ and the Cahn-Hilliard equation by J−1 ∂

∂tv for

J : W 1,20 (Ω)→W−1,2(Ω) the duality mapping, i.e. by J−1 ∂

∂tv = −Δ−1 ∂

∂tv, cf. (3.18). By using

several times Green’s formula, realize that − ∫Ω

∂∂t

v(Δ−1 ∂∂t

v) dx = ‖ ∂∂t

v‖2W−1,2(Ω)

because of

the definition (3.1) and because J−1 itself is the duality mapping W−1,2(Ω)→W 1,20 (Ω), further

that

−∫ΩΔ2v

(Δ−1 ∂v

∂t

)dx = −

∫ΩΔvΔ

(Δ−1 ∂v

∂t

)dx = −

∫ΩΔv

∂v

∂tdx =

1

2

d

dt

∫Ω|∇v|2 dx,

and∫ΩΔc(v)(Δ−1 ∂

∂tv) dx =

∫Ωc(v) ∂

∂tv dx = d

dt

∫Ωc(v) dx with c again the primitive function

to c, and also − ∫ΩΔθ(Δ−1 ∂

∂tv) dx = − ∫

Ωθ ∂∂t

v dx cancels with the corresponding term coming

from the first equation in (12.82), and obtain the estimates of θ ∈ L2(I;W 1,20 (Ω))∩L∞(I;L2(Ω))

and v ∈ L∞(I;W 1,20 (Ω))∩W 1,2(I;W−1,2(Ω)). From the first equation in (12.82), obtain now also

the estimate of ∂∂t

θ in L2(I;W−1,2(Ω)) and from the Cahn-Hilliard equation Δ2v = Δ(c(v) +

θ) − ∂∂t

v eventually the estimate of v in L2(I;W 2,2(Ω)) under a suitable qualification of c(·).Then show convergence, e.g., of Galerkin’s approximants. For details of Galerkin’s approximationwe refer (up to some sign conventions) to [230].

28Hint: Estimate the term ψ(θ)|∇v| as ∫Ωψ(θ)|∇v|v dx≤maxψ(R)2‖v‖2

L2(Ω)+ 1

4‖∇v‖2

L2(Ω;Rn)

for (12.84) or∫Ωψ(θ)|∇v| ∂

∂tv dx ≤ maxψ(R)2‖∇v‖2

L2(Ω;Rn)+ 1

4‖ ∂∂t

v‖2L2(Ω)

for (12.88).29Hint: Show the strong convergence of θ by Aubin-Lions’ Lemma 7.7 and of ∇v by uniform

monotonicity of the Laplacean as in Exercise 8.81, and then pass to the limit in the term ψ(θ)|∇v|by continuity.

30Hint: Assume �v ∈ L∞(Q;Rn) and estimate the term �v · ∇θ as∫Ω−(�v(t, ·) · ∇θ) θ dx ≤

14‖∇θ‖2

L2(Ω;Rn)+ ‖�v(t, ·)‖2

L∞(Ω;Rn)‖θ‖2

L2(Ω)for (12.83) or as

∫Ω−(�v(t, ·) · ∇θ) ∂

∂tθ dx ≤

14‖ ∂∂t

θ‖2L2(Ω)

+ ‖�v(t, ·)‖2L∞(Ω;Rn)

‖∇θ‖2L2(Ω;Rn)

for (12.91), and analogously for the term �v · ∇v

for (12.84) and (12.88). Alternatively, assuming div �v = 0 and �v|Σ = 0, use the calculations (6.33)for (12.83) and (12.84) to cause these terms to vanish.

31Hint: Show the strong convergence of θ and v by Aubin-Lions’ lemma and pass to the limitin the terms �v · ∇θ and �v · ∇v by the weak convergence.

12.6. Navier-Stokes-Nernst-Planck-Poisson-type system 421

Prigogine’s phenomenological concept [133, 340]32 balancing only the barycentricmomentum. Under certain simplifications33, it leads to a system of n+ L+ 1 dif-ferential equations combining the Navier-Stokes, and the Nernst-Planck equationmodified for moving media, and the Poisson equation for the electrostatic field:

�∂v

∂t+ �(v ·∇)v − div(μ∇v) +∇π =

L∑�=1

u�f� , div v = 0 , f� = −e�∇φ,

(12.95a)

∂u�

∂t+ div

(j� + u�v

)= r�(u1, . . . , uL) ,

j� = −d1∇u� − d2u�(e� − qtot)∇φ, = 1, . . . , L , (12.95b)

− div(ε∇φ) = qtot, qtot =L∑

�=1

e�u� , (12.95c)

with the initial conditions

v(0, ·) = v0 , u�(0, ·) = u�0 on Ω. (12.96)

The meaning of the variables is:v barycenter velocity,π pressure,u� concentration of -constituent, presumably to satisfy

∑L�=1 u� = 1, u� ≥ 0,

φ electrostatic potential,qtot the total electric charge,

and of the data is:μ > 0 viscosity,� > 0 density,e� valence (= charge) of the -constituent,ε > 0 permitivity,r�(u1, . . . , uL) production rate of the -constituent by chemical reactions,f� body force acting on the -constituent: f� = −e�∇φ, 34

j� phenomenological flux of the -constituent given in (12.95b) withd1, d2 > 0 diffusion and mobility coefficients, respectively.

32I. Prigogine received the Nobel prize in chemistry for non-equilibrium thermodynamics,particularly the theory of dissipative structures, in 1977. An alternative, Truesdell’s descriptionof mixtures, balances momenta of each constituent and counts for interactive forces among them.Both concepts, completed by energy balance, have points that are thermodynamically still notfully justified. For relations between (12.95) and Truesdel’s model see Samohyl [378].

33In particular, we consider isothermal processes, a volume additivity hypothesis with eachconstituent incompressible, small electrical currents (i.e. magnetic field is neglected), and thediffusion and mobility coefficients as well as mass densities being the same for each constituent.Moreover, we neglect the electric field outside of the specimen Ω.

34This comes from Lorenz’ force acting on a charge e� moving in the electromagnetic field(E,B), i.e. f� = e�(E + v� × B) after simplification E = −∇φ and B = 0.


In other words, j� = −d2(u�∇μ� − qtot∇φ) where

μ� = e�φ+d1d2

ln(u�), (12.97)

plays the role of an electrochemical potential. The equations (12.95a-c) thus bal-ance the barycentric momentum (under the incompressibility condition), mass ofparticular constituents, and electric induction ε∇φ. The interpretation of qtot∇φin j� is a “reaction force”35 keeping the natural requirement

∑L�=1 j� = 0 satisfied,

which eventually fixes also the constraint∑L

�=1 u� = 1. We still consider someboundary conditions, e.g. a closed container, which, in some simplified version,leads to:

v = 0, u� = u�Γ, φ = 0 on Σ. (12.98)

Besides, we naturally assume r� : RL → R continuous and the mass conservationin all chemical reactions and non-negative production rate of th constituent ifthere its concentration vanishes, and the volume-additivity constraint holds forthe initial and the boundary conditions, i.e.

L∑�=1

r�(u1, . . . , uL) = 0 , r�(u1, . . . , u�−1, 0, u�+1, . . . , uL) ≥ 0, (12.99a)

L∑�=1

u�0 = 1,

L∑�=1

u�Γ = 1, u�0 ≥ 0, u�Γ ≥ 0. (12.99b)

For analysis, we define a so-called retractK : M → {ξ∈M ; ξ� ≥ 0, = 1, . . . , L},where M denotes the affine manifold {ξ∈RL;

∑L�=1 ξ� = 1}, by

K�(ξ) :=ξ+�∑Lk=1 ξ

+k

, ξ+� := max(ξ�, 0). (12.100)

Note that K is continuous and bounded on M . Starting with u ≡ (u�)�=1,..,L and v

given such that∑L

�=1 u� = 1, we solve subsequently the Poisson, the approximateNavier-Stokes36, and finally the generalized Nernst-Planck equations, i.e.

− div(ε∇φ) = qtot , qtot =L∑

�=1

e�K�(u) , (12.101a)

�∂v

∂t+ �(v ·∇)v − div(μ∇v) +∇π = qtot∇φ , div v = 0 , (12.101b)

∂u�

∂t− div(d1∇u�) + div(u�v) = r�

(K(u)

)− div

(d2K�(u)(e� − qtot)∇φ

), = 1, .., L. (12.101c)

35This force is usually small because |qtot| is small in comparison with max�=1,...,L |e�|. Often,even the electro-neutrality assumption qtot = 0 is postulated. For derivation of this force andclarification of specific simplifications see Samohyl [378].

36The equation (12.101b) is called an Oseen problem.


Let W 1,20,div(Ω;R

n) := {v ∈ W 1,20 (Ω;Rn); div v = 0}. Recall the notion of a very

weak solution to the Navier-Stokes equation (12.95a) defined in Section 8.8.4 andanalogously to (12.101b). Also, we use this concept for (12.95b) and (12.101c). The“technical” difficulty is that the time-derivatives ∂

∂tv and ∂∂tu� are not in duality

with v and u� themselves.

Lemma 12.32 (A-priori bounds). Let (12.99) hold and let n ≤ 3. For any v ∈L2(I;W 1,2

0,div(Ω;Rn)) ∩ L∞(I;L2(Ω;Rn)) and u ∈ L2(Q;RL) such that u(·) ∈ M

a.e. in Q, the equations (12.101) have very weak solutions (v, φ, u) which satisfy:∥∥∇φ∥∥L∞(I;L2(Ω;Rn))

≤ C0, (12.102a)∥∥v∥∥L2(I;W 1,2(Ω;Rn))∩L∞(I;L2(Ω;Rn))

≤ C1, (12.102b)∥∥∥∂v∂t

∥∥∥L4/3(I;W 1,2

0,div(Ω;Rn)∗)≤ C2 + C3

∥∥v∥∥L2(I;W 1,2(Ω;R3))∩L∞(I;L2(Ω;Rn))

, (12.102c)∥∥u�

∥∥L2(I;W 1,2(Ω))∩L∞(I;L2(Ω))

≤ C4, (12.102d)∥∥∥∂u�

∂t

∥∥∥L4/3(I;W 1,2(Ω)∗)

≤ C5+C6

∥∥v∥∥L2(I;W 1,2(Ω;R3))∩L∞(I;L2(Ω;Rn))

, (12.102e)

with the constants C0, . . . , C6 independent of u and v. Besides, u satisfies theconstraint u(·) ∈ M a.e. in Q (but not necessarily u� ≥ 0).

Proof. We consider n = 3, the case n ≤ 2 being thus covered, too. Existence of veryweak solutions to the particular decoupled linear equations (12.101a), (12.101b),and (12.101c) can be proved by standard arguments, based on the bounds below.

The estimate (12.102a) is obvious if one tests (12.101a) by φ itself and realizesthat the right-hand side of (12.101a) is a-priori bounded in L∞(Q). The estimate(12.102b) for v can be obtained by testing the weak formulation of the approximateNavier-Stokes system (12.101b) by v; note that the term

∫Ω(v·∇)v · v dx vanishes,

cf. also Section 8.8.4.37

The dual estimate (12.102c) for ∂∂tv can then be obtained as in (12.51) by

testing (12.101b) by a suitable z as follows:

�∥∥∥∂v∂t

∥∥∥L4/3(I;W 1,2

0,div(Ω;Rn)∗)≤ ∥∥∇v

∥∥L2(Q;R3×3)

(μ

4√T+�N3/2

∥∥v∥∥1/2L2(I;W 1,2(Ω))

×∥∥v∥∥1/2L∞(I;L2(Ω;R3))

)+ N max

�=1,..,L|e�|

∥∥∇φ∥∥L4/3(I;L6/5(Ω))

. (12.103)

Finally, the estimate (12.102e) for u� can be obtained by testing (12.101c) by

37More precisely, we can do it in Galerkin’s approximations and then these bounds are inheritedin the limit very weak solution, too.


u�.38 The dual estimate for ∂

∂tu� can again be obtained analogously as (12.103).39

Now, we have to prove that the constraint∑L

�=1 u� = 1 is satisfied. Let us

abbreviate σ(t, ·) :=∑L�=1 u�(t, ·). By summing (12.95b) for = 1, . . . , L, one gets

∂σ

∂t=

L∑�=1

r�(K(u)) + div

(d1∇σ − vσ

+

L∑�=1

d2K�(u)(e� −

L∑k=1

ekKk(u))∇φ

)= div

(d1∇σ

)− v · ∇σ (12.104)

where (12.99a) has been used. Thus (12.104) results in the linear equation ∂∂tσ+v ·

∇σ−div(d1∇σ) = 0. We assumed σ|t=0 =∑L

�=1 u�0 = 1 and σ|Σ =∑L

�=1 u�Γ = 1on Σ, cf. (12.96) and (12.98) with (12.99b), so that the unique solution to thisequation is σ(t, ·) ≡ 1 for any t > 0.40 �

Let us abbreviate

W1 := W 1,2,4/3(I;W 1,2(Ω;RL),W 1,2(Ω;RL)∗

) ∩ L∞(I;L2(Ω;RL)), (12.105a)

W2 := W 1,2,4/3(I;W 1,2

0,div(Ω;Rn),W 1,2

0,div(Ω;Rn)∗) ∩ L∞(I;L2(Ω;Rn)). (12.105b)

If n ≤ 3, Aubin-Lions’ Lemma 7.7 gives the compact embeddings W1 �L2(I;L6−ε(Ω;RL)) for any ε > 0, and similarly W2 � L2(I;L6−ε(Ω;Rn)). More-over, let us abbreviate M : W1 × W2 ⇒ W1 × W2 defined by (u, v) ∈ M(u, v) ifu is a very weak solution to (12.101b) satisfying (12.102e) and v is a very weaksolution (12.101b) satisfying (12.102b,c) with φ a weak solution to (12.101a).41

Lemma 12.33 (Continuity). Let (12.99a) hold and let n ≤ 3. Then the set-valued mapping M is weakly* upper semicontinuous if restricted to {(u, v) ∈ W1×W2; u(·) ∈ M a.e. in Q}.

38Again, more precisely, we can do it in Galerkin’s approximations and then these bounds areinherited in the limit very weak solution, too.

39The term div(u� v) in (12.101c) suggests the estimate∫Qu�v · ∇z dx ≤ ∥∥u�‖L2(I;L6(Ω))

∥∥v∥∥1/2L2(I;L6(Ω;Rn))

∥∥v∥∥1/2L∞(I;L2(Ω;Rn))

∥∥∇z∥∥L4(I;L2(Ω;Rn))

while the last term in (12.101c) can be estimated as∫Qd2K�(u)(e� − qtot)∇φ · ∇z dxdt ≤ 2d2 max

�=1,..,L|e�|

∥∥∇φ∥∥L2(Q;Rn)

∥∥∇z∥∥L2(Q;Rn)

.

40Cf. Theorem 8.36.41Note how carefully M is defined: not every weak solution necessarily satisfies the a-priori

estimates because we cannot perform the desired tests. However, we can do it for the Galerkinsolutions and then pass to the limit so that we can show that M(u, v) is at least nonempty.


Proof. Taking a sequence of {(uk, vk)}k∈N converging weakly to (u, v) in W1×W2,by Aubin-Lions’ Lemma 7.7, uk → u strongly in L2(Q;RL), hence φk → φin Lq(I;W 1,2(Ω)), and also K�(u

k)∇φk → K�(u)∇φ in Lq(I;L2(Ω;R3)) withq < +∞ arbitrary. Then the limit passage in (12.101b) is routine; obviously∫Q(vk·∇)vk·z dxdt → ∫

Q(v·∇)v·z dxdt at least for those test functions z which

are also in L∞(Q) because vk → v strongly in L2(Q;R3) and ∇vk → ∇v weaklyL2(Q;R3×3).42 The limit passage in the very weak formulation of (12.101c) with(uk, vk, φk, uk) in place of (u, v, φ, u) easily follows by standard arguments usingthe a-priori estimates (12.102e). The a-priori estimates (12.102) themselves arepreserved in the limit, too. �

Proposition 12.34 (Existence of a fixed point). Let (12.99) hold and let n ≤ 3.The mapping (u, v) �→ (u, v) has a fixed point (u, v) on the convex set{

(u, v) ∈ W1×W2 :∥∥u∥∥

L2(I;W 1,2(Ω;RL))∩L∞(I;L2(Ω;RL))≤ C4,∥∥∥∂u

∂t

∥∥∥L4/3(I;W 1,2(Ω;RL)∗)

≤ C5+C1C3,∥∥v∥∥L2(I;W 1,2(Ω;R3))∩L∞(I;L2(Ω;R3))

≤ C1,∥∥∥∂v∂t

∥∥∥L4/3(I;W 1,2

0,div(Ω;Rn)∗)≤ C2+C1C3,

L∑�=1

u� = 1}

(12.106)

with C1, . . . , C5 from (12.102). Moreover, every such fixed point satisfies also u� ≥0 for any . Thus, considering also φ related to this fixed point (u, v), the triple(φ, v, u) is a very weak solution to the system (12.95).

Proof. The weak upper semi-continuity of M has been proved in Lemma 12.33.By a-priori estimates (12.102b-d) and by arguments such as (12.104), this map-ping maps the convex set (12.106) into itself. Both W1 and W2 are compact ifendowed with the weak* topologies. Thanks to the linearity of (12.101) and con-vexity of {(u, v)} satisfying (12.102b-d) for (u, v) given, the set M(u, v) is convex.By Lemma 12.32, also M(u, v) �= ∅. By Kakutani’s Theorem 1.11, we obtain exis-tence of a fixed point.

The constraint∑L

�=1 u� = 1 is, as proved in (12.104), satisfied and, at thisfixed point, we have additionally u�(t, ·) ≥ 0 satisfied for any t. To see this, test(12.101c) with u� = u� by the negative part u−� of u�. Realizing K�(u)∇u−� = 0because, for a.a. (t, x) ∈ Q, either K�(u(t, x)) = 0 (if u�(t, x) ≤ 0) or ∇u�(t, x)

− =0 (if u�(t, x) > 0), and r�(·)u−� ≥ 0 because of (12.99a)43, we obtain u−� = 0 a.e. inQ.

42Here we used density of L∞(Q) ∩ L2(I;W 1,20,div(Ω;Rn)) in L2(I;W 1,2

0,div(Ω;Rn)).43To be more precise, we can assume, for a moment, that r� is defined on the whole RL in

such a way that r�(u1, . . . , uL) ≥ 0 for u� < 0. As we are just proving that u� ≥ 0, the values ofr� for negative concentrations are eventually irrelevant.


The non-negativity of u� together with∑L

�=1 u� = 1 ensures that u(t, x) ∈Range(K) for a.a. (t, x) ∈ Q so that u� = K�(u) and thus the triple (φ, v, u) isa very weak solution not only to (12.101) with v = v and u = u but even to theoriginal system (12.95). �

Remark 12.35. By neglecting the Navier-Stokes part (12.95a) and considering astationary medium, i.e. v = 0 and π constant, (12.95) reduces to the Nernst-Planck-Poisson system, see Remark 12.25. Conversely, one can consider extensionof the incompressible model (12.95) for anisothermal situations, see [365, 366].44

Exercise 12.36. Prove the a-priori bounds (12.102b) and (12.102e) in detail.

Exercise 12.37. Perform the estimate (12.103) for n = 4 but in the normM (I;W 1,2

0,div(Ω;Rn)∗). 45

Exercise 12.38. Prove the limit passage in (12.101c) in detail by using the a-prioriestimates (12.102e).

Exercise 12.39 (Galerkin approach46). Apply Galerkin’s method directly to(12.95), using the retract K as in (12.101). Modify the a-priori estimates (12.102)to this case, as well as (12.104), and then make a limit passage, proving thus theexistence of the very weak solution to (12.95) without the fixed-point argument.

Exercise 12.40 (Highly viscous Stokes case). Assuming the viscous term in (12.95a)is dominant, omit the convective term �(v·∇)v in (12.95a) so that (12.95a) becomesthe Stokes equation. Modify the analysis: use a weak solution instead of the veryweak ones47 and Schauder fixed-point theorem instead of the Kakutani one.

12.7 Thermistor with eddy currents

The evolution variant of the steady-state thermistor model (6.57) should certainlycount with nonstationary heat transfer and augment (6.57a) by c(θ)∂θ∂t , cf. alsoExercises 12.45 and 12.46 below. Under higher frequencies, one should count alsowith a magnetic field induced by electric current which, when varying in time,

44This is to be done by making some parameters dependent on temperature θ, in particularthe chemical reaction rates r� = r�(u, θ), and adding the heat equation

c∂θ

∂t− div

(κ∇θ + c v θ

)= μ|∇v|2 +

L∑�=1

(f�j� + h�(θ)r�(u, θ)

)

where h�(θ) are specific enthalpies, κ the heat conductivity, and c the specific heat capacity. Thenone can show that the total energy, i.e. the sum of the kinetic, the electrostatic, the internalenergies, and the (negative) total enthalpy, i.e.

∫Ω( 1

2�|v|2 + 1

2ε|∇φ|2 + c θ −∑L

�=1 h�u�) dx, isconserved in an isolated system.

45Hint: Use z ∈ C(I;L4(Ω;R4)) in (12.52).46This more constructive approach is after [364].47Hint: Derive the “dual” estimates (12.102c,d) with L2-norms instead of L4/3-norm.

12.7. Thermistor with eddy currents 427

influences backward the electric current itself. In a so-called eddy-current approx-imation, the resulting system is48

c(θ)∂θ

∂t− div

(κ(θ)∇θ

)= σ(θ)|e|2 in Q , (12.107a)

μ0∂h

∂t+ curl e = 0 in Q , (12.107b)

curlh = σ(θ)e in Q , (12.107c)

with the following interpretation:θ is the temperature (possibly rescaled by the enthalpy transformation),e is the intensity of electric field,h is the intensity of magnetic field,σ the electric conductivity (depending on θ),c heat capacity (considered ≡ 1 without loss of generality, cf. Example 8.71),κ the heat conductivity (depending on θ), andμ0 the vacuum permeability.

The equations (12.107b,c) are the rest of the Maxwell system in electrically con-ductive medium with no magnetisation nor polarization after the mentioned eddy-current approximation. Note that one can eliminate h by applying a curl-operatorto (12.107b) and time-differentiation to (12.107c) so that, for σ constant andtaking the identity curl2h = ∇div h − Δh into account, one obtains formallyμ0σ

∂∂th −Δh = −∇div h. This reveals a parabolic character of the eddy-current

approximation of the (originally hyperbolic) full Maxwell system.49

The system (12.107) is to be completed by boundary conditions. This is alittle delicate point here, cf. also [62, 373], and we consider a simple situation inwhich there is no electro-magnetic field outside Ω and set: 50

κ(θ)∂θ

∂ν= fe and ν×h = je on Σ, (12.108a)

48Indeed, steady states of (12.107) give (6.57) because (12.107b) reduces to curl e = 0 whichmeans existence of a potential φ so that e = ∇φ, and then applying div-operator to (12.107c)yields (6.57b) because div(σ(θ)∇φ) = div(curl h− σ(θ)e) = 0.

49The full hyperbolic Maxwell system involves ε0∂e∂t

+curl h = σ(θ)e instead of (12.107c) withε0 denoting the vacuum permittivity. The eddy-current approximation neglects the so-calleddisplacement current ε0

∂e∂t

and is legitimate in highly conductive media like metals, cf. [18].50The boundary condition (12.108a) allows for injection of the normal current je like in Exer-

cise 6.30. Indeed, by using (12.107) with div curlh ≡ 0 and curl∇v ≡ 0 and assuming h and Γregular enough, we have∫

Γ(σ(θ)e·ν)v dS =

∫Γ(curl h·ν)v dS =

∫Ωcurl h·∇v + (div curlh)v dx

=

∫Ωhcurl∇v dx+

∫Γ(ν×h)·∇v dS =

∫Γ(ν×h)·∇v dS =

∫Γje·∇v dS = −

∫ΓdivS(je)v dS

for any smooth v. The last equality is due to the surface Green formula (2.103) written fora·ν = je, i.e.

∫Γ je·∇v dS =

∫Γ(divSν)(je·ν) − divS(je)v dS, when counting also that je·ν = 0.

Thus we can identify the normal electric current through Γ as je = σ(θ)e·ν = −divS (je). Forv = 1, we can also see that always

∫Γ je dS = 0, i.e. electric charge preservation.


where fe is the external heat flux and je a tangential surface current prescribedon the boundary. Further we prescribe initial conditions

θ(0, ·) = θ0 and h(0, ·) = h0 in Ω; (12.108b)

note that, in view of (12.107c), we have prescribed also e(0, ·) = curlh0/σ(θ0).

We cannot apply the transformation of the Joule heat σ|e|2 like in (6.59) but,anyhow, the problem is still well fitted to usage of a fixed-point argument aftera suitable decoupling. To this goal, we need to make the enthalpy transformationas we did in Example 8.71 and thus, up to a re-scaling of temperature and thecorresponding modification of the nonlinearities σ and κ, we can assume c constant(and equal 1 without any loss of generality). In analog to (6.61), we design thefixed point as

M := M2◦(M1×id

): ϑ �→ θ with M1 : ϑ �→ e and M2 : (ϑ, e) �→ θ, (12.109)

where, for ϑ given, e together with h forms the unique weak solution to the system

μ0∂h

∂t+ curl e = 0, curlh− σ(ϑ)e = 0 in Q, (12.110a)

ν×h = je on Σ, (12.110b)

h(0, ·) = h0 in Ω, (12.110c)

and where, for such (ϑ, e), then θ solves in a weak sense the initial-boundary-valueproblem

∂θ

∂t− div

(κ(ϑ)∇θ

)= σ(ϑ)|e|2 in Q , (12.111a)

κ(ϑ)∂θ

∂ν= fe on Σ , (12.111b)

θ(0, ·) = θ0 in Ω. (12.111c)

Obviously, any fixed point θ of M completed with the corresponding (e, h) willsolve (12.107)–(12.108) with c = 1. Note that the enthalpy transformation andthe suitable decoupling made the problem (12.111) linear so that the set of itssolution is convex. This allows for usage of Kakutani’s fixed-point Theorem 1.11without investigating a rather delicate issue of uniqueness of a solution to theheat equation (12.111) with L1-data. Thus we consider M2 and thus also M asset-valued, and assume

σ, κ : R → R continuous and bounded, inf σ(·) > 0, inf κ(·) > 0, (12.112a)

h0 ∈ L2(Ω;R3), θ0 ∈ L1(Ω), fe ∈ L1(Σ), fe ≥ 0, θ0 ≥ 0, (12.112b)

∃he ∈ W 1,1(I;L2(Ω;R3)) : je = ν×he|Σ & curlhe ∈ L2(Q;R3). (12.112c)


Note that, in fact, (12.112c) is a natural qualification of je. Then we derive thementioned existence by the following two lemmas, based on technique from [372].We denote

L2curl,0(Ω;R

3) :={v ∈ L2(Ω;R3); curl v ∈ L2(Ω;R3), v×ν = 0 on Γ

}. (12.113)

The weak formulation of (12.110) can be obtained by a curl-variant of the Greenformula51 and by the by-part integration in time, which results in∫

Q

h·curl v + e·curl z − σ(ϑ)e·v − μ0h·∂z∂t

dxdt =

∫Σ

je·v dSdt+∫Ω

μ0h0·v(0, ·) dx(12.114)

for all v, z : Q → R3 smooth with v(T, ·) = 0. The weak formulation of the originalsystem (12.107b,c) with the boundary and initial conditions (12.108) is analogous,just using θ in place of ϑ.

Lemma 12.41 (The mapping M1). For any ϑ ∈ L1(Q), the system (12.110) hasa unique weak solution (h, e) with h ∈ L∞(I;L2(Ω;R3)) and e ∈ L2(Q;R3) withcurlh ∈ L2(Q;R3), h−he ∈ L2(I;L2

curl,0(Ω;R3)), and ∂h

∂t ∈ L2(I;L2curl,0(Ω;R

3)∗).Moreover, the mapping M1 : ϑ �→ e : L1(Q) → L2(Q;R3) is continuous.

Sketch of the proof. We use the variant of (12.114) on the time interval [0, t] with-out by-part integration in time, i.e.,∫ t

0

(μ0

⟨∂h∂t

, z⟩+

∫Ω

h·curl v + e·curl z − σ(ϑ)e·v dx)dt =

∫ t

0

∫Γ

je·v dSdt,(12.115)

where 〈·, ·〉 denotes the duality between L2curl,0(Ω;R

3) and its dual. We furtherextend the identity (12.115) by continuity to allow for the test by v = −e andby z = h − he ∈ L2(I;L2

curl,0(Ω;R3)) with he from (12.112c), and use the curl-

cancellation.52 Making by-part integration, one thus obtains∫Ω

1

2|h(t)|2dx+

∫ t

0

∫Ω

σ(ϑ)

μ0|e|2dxdt =

∫ t

0

∫Ω

h·∂he

∂tdxdt−

∫Ω

he(t, ·)·h(t, ·) dx

+

∫Ω

(12h0 + he(0, ·)

)·h0 dx.

By Gronwall’s inequality, using (12.112), we obtain a-priori estimates h ∈L∞(I;L2(Ω;R3)) and e ∈ L2(Q;R3). In fact, one should execute this scenario

51This means∫Ω curlu·v dx =

∫Ω u·curl v dx +

∫Γ(u×v)·ν dS. In particular, we use∫

Ω curl h·v dx− ∫Ω h·curl v dx =

∫Γ(h×v)·ν dS =

∫Γ(ν×h)·v dS =

∫Γ je·v dS.

52The curl-cancellation means∫Ω e·curlh − h·curl e dx =

∫Γ(ν×h)·e dS, which equals zero if

the homogeneous boundary conditions ν×h = 0 are considered.


rigorously rather for some approximate solution and then pass to the limit byweak convergence to the linear system (12.110).

We then have also a-priori estimates curlh = σ(ϑ)e ∈ L2(Q;R3) and ∂h∂t ∈

L2(I;L2curl,0(Ω;R

3)∗) because 〈∂h∂t , v〉 = −〈curl e, v〉/μ0 = − ∫Qe curl v dxdt/μ0 is

bounded provided v ranges over a unit ball in L2(I;L2curl,0(Ω;R

3)). It is importantthat these estimates are uniform with respect to ϑ because of the assumption(12.112a) on σ(·). Also it is important that ∂h

∂t is in duality with h−he, and onecan thus perform the by-part integration formula in time, which yields uniquenessof the weak solution to the linear elliptic/parabolic problem (12.110).

For the claimed continuity, let us consider a sequence ϑk → ϑ and the cor-responding solutions (hk, ek) to (12.110) with ϑk in place of ϑ, then subtract thecorresponding equations and test them respectively by hk−h and ek−e. Usingagain the curl-cancellation property and the by-part integration in time, one gets

inf σ(·)∫Q

∣∣ek−e∣∣2dxdt ≤ μ0

2

∫Ω

∣∣hk(T )−h(T )∣∣2dx+

∫Q

σ(ϑk)∣∣ek−e

∣∣2dxdt=

∫Q

(σ(ϑ)−σ(ϑk)

)e·(ek−e

)dxdt → 0; (12.116)

the last convergence is due to the a-priori bounds for ek and e in L2(Ω;R3). Fromthe left-hand side of (12.116), one thus obtains the desired strong convergence. �

Lemma 12.42 (The mapping M2). For any ϑ∈L1(Q) and e∈L2(Ω;R3), the prob-lem (12.111) has a weak solution θ∈L∞(I;L1(Ω))∩Lr(I;W 1,r(Ω)) with r < 5

4 and∂θ∂t ∈ L1(I;W 1,∞(Ω)∗) and the mapping (ϑ, e) �→ {θ is a solution to (12.111)} :L1(Q)×L2(Ω;R3) ⇒ L1(Q) is upper semi-continuous and has convex non-emptyclosed values and a relatively compact range if restricted on a ball in L1(Q) witha sufficiently large radius.

Sketch of the proof. For existence of an integral solution θ ∈ C(I;L1(Ω)) see Ex-ample 9.19. In fact, it is also a weak solution with ∇θ estimated in Lr(Q;R3)with r < 5/4, cf. Remark 9.25, and ∂θ

∂t estimated in L1(I;W 1,∞(Ω)∗). As theproblem (12.111) is linear, its solutions form a convex set. This set is also closedand depends continuously on ϑ and e, as claimed. For this, we use the compactembedding Lr(I;W 1,r(Ω)) ∩ W 1,1(I;W 1,∞(Ω)∗) � L1(Q) due to Aubin-Lions’theorem as well as the continuity of the Nemytskiı mapping (ϑ, e) �→ σ(ϑ)|e|2 :L1(Q)×L2(Q;R3) → L1(Q). �

For the existence of a fixed point of M from (12.109), it then suffices touse Kakutani’s Theorem 1.11 on a sufficiently large ball in L1(Q) which M mapsinto itself. Such a ball always exists because simply the whole set M1(L

1(Ω)) isbounded in L2(Q;R3).

Exercise 12.43. Realize that σ(ϑk) → σ(ϑ) in Lp(Ω), p < +∞, but not in L∞(Ω),


and prove the convergence in (12.116).53 Realize why the other variant of (12.116)as∫Qσ(ϑ)|ek−e|2dxdt =

∫Q(σ(ϑ)−σ(ϑk))ek·(ek−e) dxdt would not give the de-

sired effect.

Exercise 12.44. Instead of the fixed-point argument (12.109), use the Rothemethod with a regularization like in (12.12), namely

θkτ−θk−1τ

τ− div

(κ(θkτ )∇θkτ

)= σ(θkτ )|ekτ |2, (12.117a)

μ0hkτ−hk−1

τ

τ+ curl ekτ = 0, (12.117b)

curlhkτ = σ(θkτ )e

kτ + τ |ekτ |η−2ekτ (12.117c)

on Ω with the boundary conditions κ(θkτ )∂∂ν θ

kτ = hk

e,τ and ν×hkτ = j ke,τ on Γ,

and with a sufficiently large η. Prove existence of a weak solution (θkτ , hkτ , e

kτ ) ∈

W 1,2(Ω) × L2(Ω;R3) × Lη(Ω;R3) such that θkτ ≥ 0, curlhkτ ∈ Lη′

(Ω;R3), andcurl ekτ ∈ L2(Ω;R3).54 Further prove the a-priori estimates from Lemmas 12.41and 12.42 together with ‖eτ‖Lη(Q;R3) = O(τ−1/η),55 and show convergence (interms of subsequences) of Rothe’s solutions towards a weak solution of the contin-uous problem.56 Alternatively to (12.117), use a semi-implicit discretization likein Remark 12.12.

Exercise 12.45. Analyse the system which arises from (12.107) by neglecting μ0∂h∂t ,

namely

c(θ)∂θ

∂t− div

(κ(θ)∇θ

)= σ(θ)|∇φ|2 on Q , (12.118a)

−div(σ(θ)∇φ

)= 0 on Q , (12.118b)

with the initial condition θ(0, ·) = θ0 and the boundary conditions

κ(θ)∂θ

∂ν= fe and σ(θ)

∂φ

∂ν= je on Σ. (12.119)

53Hint: Consider the Nemytskiı mapping ϑk �→ (σ(ϑ)−σ(ϑk))e : L1(Ω) → L2(Ω) instead ofthe mapping ϑk �→ σ(ϑk) : L

1(Ω)→ L∞(Ω).54Hint: Choose η > 4 so that, when making the test of the particular equations in (12.117)

respectively by θkτ , hkτ−h k

e,τ , and ekτ , the left-hand-side terms |θkτ |2/τ+τ |ekτ |η dominates the term

σ(θkτ )|ekτ |2θkτ arising from the right-hand side of (12.117a), so that the underlying pseudomono-tone mapping is coercive and Brezis’ Theorem 2.6 ensures existence of a solution to (12.117). Toprove non-negativity of some θkτ , test (12.117a) by (θkτ )

−.55Hint: First execute the test by 1, hk

τ−h ke,τ , and ekτ . Then test separately (12.117a) by χ(ϑk

τ ) :=

1−(1+ϑkτ )

−ε as suggested in Remark 9.25 for p = 2 to get ∇ϑτ estimated in Lr(Q;R3), r < 5/4.56Hint: The semilinear terms can be limited by weak convergence combined with Aubin-Lions

theorem. The regularizing term τ |eτ |η−2eτ can be shown to vanish in the limit. The strongconvergence of the Joule heat in L1(Q) can modify (12.116) by using also (8.52)–(8.53).


Assume the zero-current∫Γje(t, ·) dS = 0 for all t with je ∈ L1(I;L2#

′(Γ)), use

the enthalpy transformation to get c(·) = 1 like in (12.111), and prove existenceof a weak solution by modifying Lemmas 12.41 and 12.42.57

Exercise 12.46. Consider (12.118) alternatively with the boundary conditions

κ(θ)∂θ

∂ν= fe on Σ, φ|Σ

D= φ

Don ΣD,

∂φ

∂ν= 0 on ΣN, (12.120)

and transform (12.118a) into ∂θ∂t − div

(κ(θ)∇θ

)= div

(σ(θ)φ∇φ) with the bound-

ary condition κ(θ) ∂θ∂ν + σ(θ)φ∂φ∂ν = fe like in Section 6.4, referring now to the

rescaled temperature. Prove existence of a weak solution θ ∈ L2(I;W 1,2(Ω)) ∩W 1,2(I;W 1,2(Ω)∗) and φ ∈ L∞(I;W 1,2(Ω))∩L∞(Q).58 Prove uniqueness for smallfe and je.

59

12.8 Thermodynamics of magnetic materials

Magnetic materials may undergo transformation between ferromagnetic and para-magnetic states, depending mainly on temperature, and thus exhibit nontrivialcoupling with heat transfer. The resulting system, devised and analysed in [336],couples the Landau-Lifschitz-Gilbert equation (see Exercise 11.28) with the heatequation. The unknown fields are R3-valued magnetisation u and temperature θ.The resulting system is

α∂u

∂t− β(|u|)u × ∂u

∂t− μΔu+ ϕ′0(u) + θϕ′1(u) = h, (12.121a)

c(θ)∂θ

∂t− div(κ(θ)∇θ) = α

∣∣∣∂u∂t

∣∣∣2 + θϕ′1(u)·∂u

∂t(12.121b)

with the following interpretation:u magnetisation vector,θ the absolute temperature,h given outer magnetic field,α > 0 an attenuation constant,

57Hint: Design the fixed point via M1 : ϑ �→ φ and M2 : (ϑ, φ) �→ θ like before. Derive thea-priori bound of φ ∈ L1(I;W 1,2(Ω)) provided φ(t, ·) is shifted by a suitable constant, e.g. sothat

∫Ω φ(t, ·) dx = 0 for all t. Here, a version (1.58) of the Poincare inequality is to be used. For

continuity of the mapping M2 use again (12.116) modified for e = ∇φ.58Hint: Test the heat equation by θ to estimate 1

2ddt‖θ‖2

L2(Ω)+ minκ(·)‖∇θ‖2

L2(Ω;Rn)≤

maxσ(·)‖φ‖L∞(Ω)‖∇φ‖L2(Ω;Rn)‖∇θ‖L2(Ω;Rn) + ‖fe‖L2#

′(Γ)‖θ‖

L2#(Γ)by taking the a-priori

bound of φ ∈ W 1,2(Ω) ∩ L∞(Ω) into account, provided φ is shifted by a suitable constant,e.g. so that

∫Ω φ(t, ·) dx = 0 for all t. Here, a version (1.58) of the Poincare inequality is to

be used. For continuity of a fixed-point mapping or alternatively for convergence of Rothe’s orGalerkin’s approximate solutions, modify (12.116).

59Hint: Modify the procedure from Exercise 6.32.

12.8. Thermodynamics of magnetic materials 433

β = β(|u|) > 0 the inverse gyromagnetic ratio,

μ > 0 the exchange-energy constant,

κ = κ(θ) > 0 heat conductivity,

c = c(θ) > 0 heat capacity.

The thermodynamics of (12.121) can be derived from the free energy consideredpartly linearized, namely60

ψ(u, θ) = φ0(u) + θφ1(u) + φ2(θ). (12.122)

This gives the heat capacity c(θ) = −θφ′′2(θ) and allows for varying between amultiwell ψ(·, θ) for lower temperatures (corresponding to the so-called ferromag-netic phase) and a single-well ψ(·, θ) for higher temperatures (corresponding tothe so-called paramagnetic phase), cf. [336] for specific examples. Thus “continu-ous switching” of the multi/single-well character of ψ(·, θ) model the ferro/para-magnetic transformation. In view of (12.127b,d) below, the slight growth of bothc(θ) and κ(θ) like ∼ θ1/5+ε is needed for the analysis, similarly as in (12.14a,b).

To facilitate its analysis, we consider it after the enthalpy transformation anddefine

γ(u, ϑ) = θϕ′1(u), κ(ϑ) =κ(θ)

c(θ)with θ = c−1(ϑ) where c(θ) =

∫ θ

0

c(r) dr.

(12.123)

In terms of the re-scaled temperature ϑ = c(θ), (12.121) takes the form:

α∂u

∂t− β(|u|)u× ∂u

∂t− μΔu+ ϕ′0(u) + γ(u, ϑ) = h, (12.124a)

∂ϑ

∂t− div(κ(ϑ)∇ϑ) = α

∣∣∣∂u∂t

∣∣∣2 + γ(u, ϑ)·∂u∂t

(12.124b)

on Q. We complete (12.124) by the initial conditions

u(0, ·) = u0, ϑ(0, ·) = ϑ0 in Ω, (12.125a)

and the boundary conditions (for simplicity linear in terms of ϑ):

∂u

∂ν= 0, κ(ϑ)

∂ϑ

∂ν+ bϑ = bϑe on Σ. (12.125b)

It is illustrative to prove existence of a weak solution to the initial-boundary-value-problem (12.124) by a suitable regularization. For ε > 0, we add an attenu-

60Note that, in terms of ψ, the system (12.121) reads as α ∂u∂t−β(|u|)u× ∂u

∂t−μΔu+ψ′

u(u, θ) =

h and θ ∂∂t

ψ′θ(u, θ)+div(κ(θ)∇θ) = −α| ∂u

∂t|2. The latter equation is, in fact, the entropy equation.


ation ε|∂u∂t |p−2 ∂u∂t with a sufficiently large p. Thus we consider the system(

α+ε∣∣∂uε

∂t

∣∣p−2)∂uε

∂t− β(|uε|)uε×∂uε

∂t− μΔuε + ϕ′0(uε) + γ(uε, ϑε) = h,

(12.126a)

∂ϑε

∂t− div

(κ(ϑε)∇ϑε

)= α

∣∣∣∂uε

∂t

∣∣∣2 + γ(uε, ϑε)·∂uε

∂t(12.126b)

on Q with the partly regularized initial and boundary conditions

uε(0, ·) = u0, ϑε + uε(0, ·) = ϑ0,ε in Ω, (12.126c)

∂uε

∂ν= 0, κ(ϑε)

∂ϑε

∂ν+ bϑε = bϑe,ε on Σ. (12.126d)

We make the following assumptions:

β, κ : R+ → R+, ϕ′0 : R3 → R3, γ : R3×R+ → R3 continuous, (12.127a)

∃ ε>0, C∈R ∀(u, ϑ)∈R3×R+ : |γ(u, ϑ)| ≤ C(1+|ϑ|5/6−ε), (12.127b)

ϕ0(u) ≥ ε|u|2, |ϕ′0(u)| ≤ C(1+|u|2), (12.127c)

κ(ϑ) ≥ ε, κ(ϑ) ≤ C, |β(|u|)| ≤ C(1+|u|2), (12.127d)

b∈L∞(Σ), b ≥ 0, ϑe∈L1(Σ), ϑe ≥ 0, h∈W 1,1(I;L2(Ω;R3)), (12.127e)

u0 ∈ W 1,2(Ω;R3), ϑ0 ∈ L1(Ω), ϑ0 ≥ 0. (12.127f)

Lemma 12.47. Let p ≥ 12 and ε > 0 be fixed, and let ϑe,ε∈L2(Σ) and ϑ0,ε∈L2(Ω).The regularized system (12.126) possesses a weak solution uε ∈ W 1,p(I;Lp(Ω;R3))and ϑε ∈ L2(I;W 1,2(Ω)) ∩W 1,2(I;L2(Ω)) such that ϑε ≥ 0.

Sketch of the proof. Realize the structure of (12.126) as the pseudo-parabolic equa-tion similarly like in Exercise 11.28. Besides convergence of some approximate so-lutions (cf. Exercise 12.51 below), the essential point is to execute suitable a-prioriestimates. For this, test (12.126a) by ∂uε

∂t and (12.126b) by ϑε. Sum it up to obtain

d

dt

∫Ω

μ

2

∣∣∇uε

∣∣2 + ϕ0(uε) +1

2|ϑε|2 dx

+

∫Ω

α∣∣∣∂uε

∂t

∣∣∣2 + ε∣∣∣∂uε

∂t

∣∣∣p + κ(ϑε)|∇ϑε|2 dx+

∫Γ

b ϑ2ε dS

=

∫Ω

α∣∣∣∂uε

∂t

∣∣∣2ϑε +(ϑε−1

)γ(uε, ϑε)·∂uε

∂t+ h· ∂uε

∂tdx+

∫Γ

b ϑe,εϑε dS

≤ α∥∥∥∂uε

∂t

∥∥∥2L4(Ω;R3)

∥∥ϑε

∥∥L2(Ω)

+ C(1 +

∥∥ϑε

∥∥11/6−ε

L2(Ω)

)∥∥∥∂uε

∂t

∥∥∥L4(Ω;R3)

+∥∥h∥∥

L2(Ω;R3)

∥∥∥∂uε

∂t

∥∥∥L2(Ω;R3)

+∥∥b∥∥

L∞(Γ)

∥∥ϑe,ε

∥∥L2(Γ)

∥∥ϑε

∥∥L2(Γ)

(12.128)


with a sufficiently large C. Note that the orthogonality (β(|uε|)uε×∂uε

∂t )·∂uε

∂t = 0has been used. By Young’s and Gronwall’s inequalities, one obtains the estimatesuε ∈ W 1,p(I;Lp(Ω;R3)) and ϑε ∈ L2(I;W 1,2(Ω))∩L∞(I;L2(Ω)); here p ≥ 12 and

then the estimate ‖ϑε‖11/6−εL2(Ω) ‖∂uε

∂t ‖L4(Ω;R3) ≤ Cδ + δ‖ϑε‖2L2(Ω) + δ‖∂uε

∂t ‖12L12(Ω;R3)

with a suitably small δ > 0 has been used.Moreover, test (12.126b) by ∂ϑε

∂t to get ∂ϑε

∂t ∈ L2(Q) when realizing that the

right-hand side of (12.126b) is in L2(Q) since ∂uε

∂t ∈ L12(Q;R3) and ϑε ∈ L10/3(Q)have already been proved.

Eventually, ϑε ≥ 0 can be obtained by testing (12.126b) by ϑ−ε , exploitingthat γ(u, 0) = 0 by definition of the nonlinearity γ; here we can, for a moment,assume γ(u, ·) = 0 defined for negative arguments. �

Further, relying on ϑε ≥ 0, physically motivated estimates are to be derived.

Lemma 12.48. Let the assumptions of Lemma 12.47 be fulfilled and, in addition,also ‖ϑe,ε‖L1(Σ) and ‖ϑ0,ε‖L1(Ω) be bounded independently of ε. Then, with C andCr independent of ε > 0, it holds that∥∥uε

∥∥L∞(I;W 1,2(Ω;R3))∩W 1,2(I;L2(Ω;R3))

≤ C, (12.129a)∥∥ϑε

∥∥L∞(I;L1(Ω))∩W 1,1(I;W 3,2(Ω)∗) ≤ C, (12.129b)∥∥∇ϑε

∥∥Lr(Q;R3)

≤ Cr, 1 ≤ r < 5/4, (12.129c)∥∥∥∂uε

∂t

∥∥∥Lp(Q;R3)

≤ C p√1/ε. (12.129d)

Sketch of the proof. First, test (12.126b) by 1 and (12.126a) again by ∂uε

∂t . We will

see cancellation effects of±α|∂uε

∂t |2 and±γ(uε, ϑε)· ∂uε

∂t and again the orthogonality

(β(|uε|)uε×∂uε

∂t )·∂uε

∂t = 0, and obtain the energy balance:

d

dt

∫Ω

μ

2

∣∣∇uε

∣∣2 + ϕ0(uε) + ϑε dx

+

∫Γ

b ϑεdS +

∫Ω

ε∣∣∣∂uε

∂t

∣∣∣p dx =

∫Γ

b ϑe dS +

∫Ω

h· ∂uε

∂tdx. (12.130)

Then we integrate it over [0, t] and make the by-part integration for the lastterm. Then (12.129a,d) and also the first part of (12.129b) follow by Gronwall’sinequality.

Further, we test (12.126b) by χ(ϑε) = 1 − (1+ϑε)δ with δ > 0 and sum

it with (12.126a) tested by k ∂uε

∂t with a sufficiently large k, and proceed like in(12.17)–(12.24) by using also (12.127b). Thus (12.129c) follows.

Eventually, testing (12.126b) by v ∈ L∞(I;W 3,2(Ω)), we obtain the secondpart of (12.129b). �Proposition 12.49 (Convergence with ε → 0). Let, in addition to the assumptionsof Lemmas 12.47 and 12.48, also ϑe,ε → ϑe in L1(Σ) and ϑ0,ε → ϑ0 in L1(Ω).


There is a subsequence {(uε, ϑε)}ε>0 converging weakly* in the topology indicatedin the estimates (12.129a-c) to some (u, ϑ) and any (u, ϑ) obtained by this way isa weak solution to the initial-boundary-value problem (12.124)–(12.125).

Proof. The weakly* converging subsequence is to be selected by Banach’s Theo-rem 1.7. By Aubin-Lions’ theorem with interpolation, we have also strong conver-gence of ϑε → ϑ in L5/3−ε(Q). By the Sobolev embedding W 1,2(Q) � L4−ε(Q), wehave uε → u in L4−ε(Q;R3). The convergence in the “asymptotically semilinear”equation (12.126a) to (12.124a) is then by the weak convergence; more in detail,

the only quasilinear term ε∣∣∂uε

∂t

∣∣p−2 ∂uε

∂t is in (12.126a) but it disappears in thelimit due to the estimate∣∣∣∣ ∫

Q

ε∣∣∣∂uε

∂t

∣∣∣p−2 ∂uε

∂t·v dxdt

∣∣∣∣ ≤ ε∥∥∥∂uε

∂t

∥∥∥p−1

Lp(Q;R3)

∥∥v∥∥Lp(Q;R3)

= O(ε1/p) → 0

for any v smooth, cf. (12.129d).

Testing (12.124a) by ∂u∂t gives the magnetic-energy balance∫

Ω

ϕ0

(u(T )

)+

μ

2|∇u(T )|2dx+

∫Q

α∣∣∣∂u∂t

∣∣∣2dxdt=

∫Ω

ϕ0

(u0) +

μ

2|∇u0|2dx+

∫Q

(h− γ(u, ϑ)

)·∂u∂t

. (12.131)

To execute rigorously the proof of (12.131), one needs the formulas∫Q

ϕ′0(u)·∂u

∂tdxdt =

∫Ω

ϕ0

(u(T )

)− ϕ0

(u0) dx, and (12.132a)∫

Q

μΔu·∂u∂t

dxdt =μ

2

∫Ω

|∇u0|2 − |∇u(T )|2 dx. (12.132b)

For (12.132a), we use that ∂u∂t ∈ L2(Q;R3) and ϕ′0(u) ∈ L2(Q;R3) due to

(12.127c) since u ∈ L4(Q;R3) has been proved. Since it has also been provedthat ∂u

∂t ∈L2(Q;R3), it follows from (12.124a) that, in addition, Δu∈L2(Q;R3).

Here we also use that we have ϑ∈L5/3−ε(Q) and, by (12.127b), we can see thatγ(u, ϑ)·∂u∂t ∈L2(Q;R3). Moreover, u : I → W 1,2(Ω;R3) is actually a weakly contin-uous function (although not necessarily strongly continuous). Thus the integration-by-parts formula (12.132b) has indeed a good sense. 61

61More in detail, (12.132b) can be proved by mollifying u with respect to the spatial variables,not with respect to time as used in Lemma 7.3 through Lemma 7.2: Denoting by Mη the mol-

lification operator, for uη := Mηu we have∂uη

∂t= Mη

∂u∂t∈ L2(I;C1(Ω;R3)). By standard cal-

culus, we obtain that (12.132b) holds for the mollified function uη , namely:∫QΔuη · ∂uη

∂tdxdt =

12

∫Ω|∇(Mηu0)|2−|∇uη(T )|2 dx. We then obtain (12.132b) by letting η → 0, using the fact that

Δuη → Δu in L2(Q;R3),∂uη

∂t→ ∂u

∂tin L2(Q;R3), and ∇uη(T )→ ∇u(T ) in L2(Ω;R3×3).


Then the strong convergence ∂uε

∂t → ∂u∂t in L2(Q) can be proved by∫

Q

α∣∣∣∂u∂t

∣∣∣2dxdt ≤ lim infε→0

∫Q

α∣∣∣∂uε

∂t

∣∣∣2dxdt ≤ lim supε→0

∫Q

α∣∣∣∂uε

∂t

∣∣∣2dxdt= lim sup

ε→0

∫Ω

μ

2|∇u0|2 + ϕ0(u0)− μ

2|∇uε(T )|2 − ϕ0(uε(T )) dx

+

∫Q

(h− γ(uε, ϑε)

)·∂uε

∂tdxdt

≤∫Ω

μ

2|∇u0|2 + ϕ0(u0)− μ

2|∇u(T )|2 − ϕ0(u(T )) dx

+

∫Q

(h− γ(u, ϑ)

)·∂u∂t

dxdt =

∫Q

α∣∣∣∂u∂t

∣∣∣2dxdt, (12.133)

where the weak lower semicontinuity of the convex functionals u �→ ∫Q α|u|2 dxdt

and u �→ ∫Ω

μ2 |∇u|2dx, the equation (12.126a), and the strong convergence

uε(T ) → u(T ) in L5(Ω;R3) and γ(uε, ϑε) → γ(u, ϑ) in L2(Q;R3) have succes-sively been used, and eventually also (12.131) has been used for the last equalityin (12.133). Thus, in fact, we have equalities everywhere in (12.133) and, in partic-ular, limε→0

∫Q α|∂uε

∂t |2dxdt =∫Q α|∂u∂t |2dxdt. The weak convergence ∂uε

∂t → ∂u∂t in

L2(Q;R3) is thus converted to the strong convergence; cf. Exercise 2.52. Then thelimit passage in the heat-transfer equation (12.126b) towards (12.124b) is easy. �Remark 12.50 (Ferromagnets with eddy currents). By combination of (12.121)with (12.107), one obtains the system62

α∂u

∂t− β(|u|)u× ∂u

∂t− μΔu+ ϕ′0(u) + θϕ′1(u) = μ0h, (12.134a)

c(θ)∂θ

∂t− div

(κ(θ)∇θ

)= α

∣∣∣∂u∂t

∣∣∣2 + θϕ′1(u)·∂u

∂t+ σ(θ)|e|2, (12.134b)

μ0∂h

∂t+ curl e = −μ0

∂u

∂t, (12.134c)

curlh = σ(θ)e. (12.134d)

Note that the right-hand sides of (12.134a) and of (12.134c) are now not given,in contrast to (12.121a) and (12.107b). This system describes thermodynamics ofelectrically conductive ferromagnets under a possible ferro/para magnetic trans-formation with Joule heating by the induced electric current governed by theMaxwell system considered in the eddy-current approximation.

Exercise 12.51. Apply Galerkin method to (12.126) and, using a-priori estimatesof Lemma 12.47, show convergence of the Galerkin solutions.63

62For such a system even augmented by a mechanical part describing magnetostrictive mate-rials under small strains see [373].

63Hint: Show strong convergence of ∂∂t

uεk → ∂∂t

uε in Lp(Q;R3) for k → ∞ (k indexing the

Galerkin approximants) by using the d-monotonicity of the regularizing term ε| ∂∂t

uεk|p−2 ∂∂t

uεk.


Exercise 12.52. Considering an initial-boundary-value problem for the system(12.134), derive the a-priori estimates and prove existence of a weak solution.64

12.9 Thermo-visco-elasticity: fully nonlinear theory

The last example of coupled nonlinear systems refines the thermo-visco-elasticityfrom Section 12.1 where we used a partly linear ansatz (12.1) as ψ′′′eθθ ≡ 0 and thusthe heat capacity c = −θψ′′θθ(e, θ) was independent of θ. Now we consider the fullynonlinear specific Helmholtz free energy. This will require us also to use a gradienttheory for e. Referring to the notation of Section 12.1, we use the free energy

ψ1(e, θ,∇e) := ψ(e, θ) +γ

2|∇e|2 (12.135)

and the dissipation rate

ξ1(e,∇e) := ξ(e) + γ1|∇e|2. (12.136)

Obviously, for γ = 0 = γ1 and for ψ from (12.1) and ξ from (12.2), we obtain themodel from Section 12.1.

For simplicity, like in (12.2), we will consider ξ quadratic so that ξ1 is, up tothe factor 1

2 , simultaneously the (pseudo)potential of the dissipative forces.The force balance now reads as

�∂2u

∂t2− div

(σ − div h

)= g (12.137)

with the stress σ = σ(e,∇e, e,∇e, θ) := [ψ1]′e(e, θ,∇e) + 1

2 [ξ1]′e(e,∇e) and the

so-called hyperstress h = h(e,∇e, e,∇e, θ) := [ψ1]′∇e(e, θ,∇e) + 1

2 [ξ1]′∇e(e,∇e).65

Thus, in view of the ansatz (12.135) and (12.136), we have simply σ = σ(e, e, θ) =ψ′e(e, θ) +

12ξ′e and h = h(∇e,∇e) = γ∇e + γ1∇e, and thus we consider (12.137)

in the specific form:

�∂2u

∂t2− div

(12ξ′e(∂u∂t

)+ ψ′e(e(u), θ)−Δe

(γ1

∂u

∂t+γu

))= g. (12.138a)

As in Section 12.1, the entropy is defined by s = −[ψ1]′θ(e, θ,∇e) = −ψ′θ(e, θ) and

the entropy equation reads as θ ∂s∂t = ξ1(e(

∂u∂t ),∇e(∂u∂t )) + div j. Considering an

anisotropic nonlinear Fourier law for the heat flux j = K(e(u), θ)∇θ, we obtainthe heat equation

c(e(u), θ)∂θ

∂t− div(K(e(u), θ)∇θ) = ξ

(e(∂u∂t

))+ γ1

∣∣∣∇e(∂u∂t

)∣∣∣2+ θψ′′eθ

(e(u), θ

):e(∂u∂t

)with c(e, θ) = −θψ′′θθ(e, θ). (12.138b)

64Hint: Apply enthalpy transformation, and use the test from Section 12.7, noting also thecancellation of the terms ±μ0

∂u∂t·h.

65The concept of hyperstresses [338] is related with so-called 2nd-grade nonsimple materials(also called multipolar solids or complex materials), cf. e.g. [153, 385].

12.9. Thermo-visco-elasticity: fully nonlinear theory 439

We consider again the initial conditions (12.6b). We will use natural Newton-typeboundary conditions, which means66(1

2ξ′e(∂u∂t

)+ ψ′e(e(u), θ)−Δe

(γ1

∂u

∂t+γu

))·ν − divS

(∇e(γ1

∂u

∂t+γu

)·ν) = h,

(12.139a)

∇e(u):(ν⊗ν) = 0 , and (12.139b)

K(e(u), θ)∂θ

∂ν= f on Σ; (12.139c)

cf. (2.107). In (12.139a), h stands for the prescribed true boundary force. Thisleads to the weak formulation of the mechanical part (12.138a)-(12.139a,b) as∫

Q

(12ξ′e(∂u∂t

)+ ψ′e(e(u), θ)

):e(v) +∇e

(γ1

∂u

∂t+γu

).:∇e(v)− �

∂u

∂t·∂v∂t

dxdt

+

∫Ω

�∂u

∂t(T )·v(T ) dx =

∫Ω

�v0·v(T ) dx+

∫Q

g·v dxdt+∫Σ

h·v dSdt (12.140)

for any v ∈ W 1,2(I;L2(Ω;Rn)) ∩ L2(I;W 2,2(Ω;Rn)); naturally, “.: ” means sum-

mation over three indices.The peculiarity is that the heat capacity c depends also on the mechani-

cal variable e and one cannot perform the conventional enthalpy transformation(8.198). Anyhow, one can make an enhanced enthalpy transformation, which re-quires gradient theory for e. This procedure is based on an elementary calculus:

c(e, θ)∂θ

∂t=

∂ c (e, θ)

∂t− c ′e(e, θ):

∂e

∂t(12.141)

where

c (e, θ) :=

∫ θ

0

c(e, t) dt , and thus c ′e(e, θ) =∫ θ

0

c′e(e, t) dt. (12.142)

We introduce the substitution

ϑ = c (e(u), θ). (12.143)

It is physically natural to assume the heat capacity c positive, which makes c (e, ·)increasing and thus also invertible, which allows us to define

Θ(e, ϑ) :=[c (e, ·)]−1

(ϑ), (12.144a)

K0(e, ϑ) := K(e,Θ(e, ϑ))Θ′ϑ(e, ϑ), (12.144b)

K1(e, ϑ) := K(e,Θ(e, ϑ))Θ′e(e, ϑ), (12.144c)

F (e, ϑ) := Θ(e, ϑ)ψ′′θe(e,Θ(e, ϑ)) + c ′e(e,Θ(e, ϑ)). (12.144d)

66We use twice the Green formula∫Ω divΔe(u)·z dx =

∫Γ Δe(u):(z⊗ν) dS−∫

Ω Δe(u):∇z dx =∫Ω∇e(u)

.:∇2z dx+

∫Γ Δe(u):(z⊗ν) − ∇e(u)

.:(∇z⊗ν) dS =

∫Ω∇e(u)

.:∇e(z) +

∫Γ Δe(u):(z⊗ν) −

∇e(u).:(( ∂z

∂νν+∇Sz)⊗ν) dS together with the formula (2.104). Note that, in (12.139a), one ex-

pects the term(divSν

)∇e(γ1

∂u∂t

+γu):(ν⊗ν) which, however, vanishes because of (12.139b).


Realizing θ = Θ(e(u), ϑ), one has

K(e(u), θ)∇θ = K(e(u),Θ(e(u), ϑ))∇Θ(e(u), ϑ)

= K0(e(u), ϑ)∇ϑ + K1(e(u), ϑ)∇e(u). (12.145)

Also, we have F (e, ϑ) = F(e, θ) := θψ′′eθ(e, θ) +∫ θ

0 c′e(e, t) dt for θ = Θ(e, ϑ) sothat, since c′e(e, θ) = −θψ′′′eθθ(e, θ), we have

[F−ψ′e◦Θ

](e, ϑ) =

[F−ψ′e

](e, θ) = θψ′′eθ(e, θ)−

∫ θ

0

tψ′′′eθθ(e, t) dt− ψ′e(e, θ).

(12.146)

Furthermore,[F−ψ′e

]′θ(e, θ) = θψ′′′eθθ(e, θ) + ψ′′eθ(e, θ)− θψ′′′eθθ(e, θ)− ψ′′eθ(e, θ) = 0

so that F−ψ′e◦Θ is independent of ϑ. On the other hand, from (12.146), one cansee that [F−ψ′e](e, 0) = −ψ′e(e, 0). Therefore, we can deduce that[

ψ′e◦Θ− F](e, ϑ) = ψ′e(e, 0) = ϕ′(e), (12.147)

where we again abbreviated ϕ(·) := ψ(·, 0) as in Section 12.1. This transforms thesystem (12.138) into the form

�∂2u

∂t2− div

(12ξ′e(

∂u

∂t) + ϕ′(e(u)) + F (e(u), ϑ) −Δe

(γ1

∂u

∂t+γu

))= g, (12.148a)

∂ϑ

∂t− div

(K0(e(u), ϑ)∇ϑ

)= ξ

(e(∂u∂t

))+ γ1

∣∣∣∇e(∂u∂t

)∣∣∣2+ F

(e(u), ϑ

):e(∂u∂t

)+ div

(K1(e(u), ϑ)∇e(u)

), (12.148b)

with the boundary conditions(12ξ′e(∂u∂t

)+ ϕ′(e(u)) + F (e(u), ϑ)−Δe

(γ1

∂u

∂t+γu

))·ν− divS

(∇e(γ1

∂u

∂t+γu

)·ν) = h, (12.149a)

∇e(u):(ν⊗ν) = 0 , and (12.149b)

K0(e(u), ϑ)∂ϑ

∂ν+ K1(e(u), ϑ)

∂e(u)

∂ν= f on Σ, (12.149c)

and the initial conditions

u(0, ·) = u0,∂u

∂t(0, ·) = v0, ϑ(0, ·) = c (e(u0), θ0) on Ω. (12.149d)


We analyse the system (12.148) by Rothe’s method with a suitable regular-ization to compensate the growth of the non-monotone terms on the right-handside of the heat equation (12.148b). More specifically, we consider

�ukτ−2uk−1

τ +uk−2τ

τ2− div

(1

2ξ′e(uk

τ−uk−1τ

τ

)+ ϕ′

(e(uk

τ ))

+ F (e(ukτ ), ϑ

kτ ) + τ

∣∣e(ukτ )∣∣η−2

e(ukτ )− div hkτ

)= gkτ

with hkτ = ∇e(γ1

ukτ−uk−1

τ

τ+γuk

τ

)+ τ

∣∣∇e(ukτ )∣∣η−2∇e(uk

τ ), (12.150a)

ϑkτ−ϑk−1

τ

τ− div

(K0(e(u

kτ ), ϑ

kτ )∇ϑk

τ

)=(1−

√τ

2

)ξ(e(uk

τ−uk−1τ

τ

))+ γ1

∣∣∣∇e(uk

τ−uk−1τ

τ

)∣∣∣2 + F(e(uk

τ ), ϑkτ

):e(uk

τ−uk−1τ

τ

)+ div

(K1(e(u

kτ ), ϑ

kτ )∇e(uk

τ )). (12.150b)

We complete this system with the boundary conditions(12ξ′e(uk

τ−uk−1τ

τ

)+ ϕ′

(e(uk

τ ))+ F

(e(uk

τ), ϑkτ ))

+ τ∣∣e(uk

τ )∣∣η−2

e(ukτ )− div hkτ

)·ν − divS

(hkτ ·ν

)= hk

τ , (12.151a)

hkτ :(ν⊗ν) = 0 , and (12.151b)

K0(e(ukτ ), ϑ

kτ )

∂ϑkτ

∂ν+ K1(e(u

kτ ), ϑ

kτ )

∂e(ukτ )

∂ν= fk

τ on Γ. (12.151c)

We start this recursive scheme for k = 1 by considering the initial conditions

u0τ = u0τ , u−1

τ = u0τ − τv0, ϑ0τ = c (e(u0τ ), θ0) on Ω (12.151d)

involving a suitably regularized initial displacement u0τ .The involved higher gradients allow for a general free energy ψ(e, θ). Re-

call that the function ϕ is called semi-convex if ϕ(·) +K| · |2 is convex for somesufficiently large K. In terms of the transformed data, we assume:

ϕ(·) = ψ(·, 0) : Rn×nsym → R+ is semi-convex and (12.152a)

∃C ∈ R : ϕ(·) ≤ C(1+| · |2∗), ∣∣ϕ′(·)∣∣ ≤ C

(1+| · |2∗−1−ε

), (12.152b)

∀e ∈ Rn×nsym : ϕ(e):e+ C ≥ 0, (12.152c)

∃CK0, CK1

∈ R, κ0, ε > 0 ∀ (e, ϑ) ∈ Rn×nsym ×R, s ∈ Rn :

K0(e, ϑ)s·s ≥ κ0|s|2, (12.152d)∣∣K0(e, ϑ)∣∣ ≤ CK0

(1+|e|2∗/(n+2)−ε+|ϑ|1/n−ε

), (12.152e)


∣∣K1(e, ϑ)∣∣ ≤ CK1

√1+|ϑ|, (12.152f)

∃CF∈ R, 0 ≤ p

F< 2∗/2, 0 ≤ q

F< 1 ∀ (e, ϑ) ∈ Rn×n

sym ×R : (12.152g)∣∣F (e, ϑ)∣∣ ≤ CF

(1+|e|pF +|ϑ|qF

), (12.152h)

�, γ1, γ > 0, ξ : Rn×nsym →R quadratic, coercive, (12.152i)

g ∈ L2(I;L2∗′(Ω;Rn)), h ∈ L2(I;L2#

′(Γ;Rn)), (12.152j)

u0 ∈ W 2,2(Ω;Rn), v0 ∈ L2(Ω;Rn), (12.152k)

θ0 ≥ 0, f ≥ 0, c (θ0, e(u0)) ∈ L1(Ω), f ∈ L1(Σ). (12.152l)

In fact, we will need (12.152) only for ϑ ≥ 0 in what follows. The semiconvexityof ϕ assumed in (12.152a) is a rather technical assumption facilitating a simpleusage of Rothe’s method, otherwise the ψ′e-term is in a position of the lower-order term and its monotonicity is not essentially needed. Note that (12.152e)ensures integrability of K0(e(u), ϑ)∇ϑ if e(u) ∈ L2∗(Q;Rn×n), ϑ ∈ L(n+2)/n−ε(Q),and ∇ϑ ∈ L(n+2)/(n+1)−ε(Q;Rn). Similarly, also (12.152f) ensures integrability ofK1(e(u), ϑ)∇e(u).

Lemma 12.53 (Existence of Rothe’s solution). Let (12.152) hold and letu0τ ∈ W 2,η(Ω;Rn) with η > max(4, 2pF+2, 2/(1−qF )). Then the recursiveboundary-value problem (12.150)–(12.151) possesses a weak solution (uk

τ , ϑkτ ) ∈

W 2,η(Ω;Rn)×W 1,2(Ω) such that ϑkτ ≥ 0 for any k = 1, ..., T/τ .

Proof. To prove the coercivity of the underlying nonlinear operator, the particularequations in (12.150) are to be tested respectively by uk

τ and ϑkτ , and the non-

monotone terms are to be estimated by Holder’s and Young’s inequalities. Let usbriefly discuss the estimation of the nonmonotone terms under this test. Using(12.152f), one can estimate∣∣∣∣ ∫

Ω

K1(e(ukτ ), ϑ

kτ )∇e(uk

τ )·∇ϑkτ dx

∣∣∣∣ ≤ ∫Ω

CK1

√1+|ϑk

τ |∣∣∇e(uk

τ )∣∣ ∣∣∇ϑk

τ

∣∣ dx≤∫Ω

CK1

(1+√|ϑk

τ |)∣∣∇e(uk

τ )∣∣ ∣∣∇ϑk

τ

∣∣dx≤ Cδ,η + δ

∥∥√|ϑkτ |∥∥4L4(Ω)

+ δ∥∥∇e(uk

τ )∥∥ηLη(Ω;Rn×n×n)

+ δ∥∥∇ϑk

τ

∥∥2L2(Ω;Rn)

(12.153)

with δ > 0 arbitrarily small and some Cδ,η depending on η and δ; here we usedthat η > 4. The heat sources with quadratic growth can be estimated as∣∣∣∣ ∫

Ω

(ξ(e(uk

τ ))+γ1|∇e(ukτ )|2

)ϑkτ dx

∣∣∣∣ ≤ Cδ,η + δ∥∥e(uk

τ )∥∥ηLη(Ω;Rn×n)

+ δ∥∥∇e(uk

τ )∥∥ηLη(Ω;Rn×n×n)

+ δ∥∥ϑk

τ

∥∥2L2(Ω)

; (12.154)

here we again used that η > 4. By (12.152h), the adiabatic term in (12.150b) can


be estimated as∣∣∣∣ ∫Ω

F (e(ukτ ), ϑ

kτ ):e(u

kτ)ϑ

kτ dx

∣∣∣∣ ≤ CF

∫Ω

(1+|e(uk

τ )|pF +|ϑkτ |qF

)∣∣e(ukτ )∣∣ ∣∣ϑk

τ

∣∣dx≤ C′

F+∥∥e(uk

τ )∥∥ηLη(Ω;Rn×n)

+ δ‖ϑkτ‖2L2(Ω) (12.155)

with C′F

dependent on δ and on CF , pF , and qF from (12.152h); here we used thatη > max(2pF+2, 2/(1−qF )).

The pseudomonotonicity of this operator is obvious because all non-monotoneterms are either in a position of lower-order terms with the exception of the K1-term in (12.150b). This term is however linear in ∇e(uk

τ ), hence weakly continuous.Then the claimed existence follows by using Brezis’ Theorem 2.6.

Eventually, one can test (12.150b) by (ϑkτ )−, which reveals ϑk

τ ≥ 0 whenrealizing that, for a moment, one can define F (e, ϑ) = 0 for ϑ ≤ 0, and then foundthat this possible re-definition is, in fact, irrelevant for this particular solution. �

Lemma 12.54 (Energy estimates). Let the assumptions of Lemma 12.53 hold,and let also ‖u0τ‖W 2,η(Ω;Rn) = O(τ−1/η). Then:∥∥uτ

∥∥W 1,∞(I;L2(Ω;Rn))∩L∞(I;W 2,2(Ω;Rn))

≤ C, (12.156a)∥∥ϑτ

∥∥L∞(I;L1(Ω))

≤ C, (12.156b)∥∥e(uτ )∥∥L∞(I;W 1,η(Ω;Rn×n))

≤ Cτ−1/η. (12.156c)

Sketch of the proof. First we test the mechanical part (12.150a) by (ukτ−uk−1

τ )/τ .We use convexity of the kinetic energy �

2 | · |2, of E �→ γ2 |E|2 + τ

η |E|η, and of the

regularizing functional τη | · |η. We further use the semi-convexity of ϕ and the

coercivity of the quadratic form ξ to estimate, as in (8.86),

(ϕ′(e(uk

τ )) +1

2ξ′e(uk

τ − uk−1τ

τ

)):e(uk

τ − uk−1τ

τ

)=(ϕ′(e(uk

τ )) +1

2√τξ′e(uk

τ )):e(uk

τ − uk−1τ

τ

)− ξ′e(uk−1

τ )

2√τ

:e(uk

τ − uk−1τ

τ

)+ (1−√

τ )ξ(e(ukτ−uk−1

τ

τ

))≥ 1

τ

(ϕ(e(uk

τ )) +ξ(e(uk

τ ))

2√τ

− ϕ(e(uk−1τ ))− ξ(e(uk−1

τ ))

2√τ

)− ξ′e(uk−1

τ )

2√τ

:e(uk

τ−uk−1τ

τ

)+ (1−√

τ )ξ(e(ukτ−uk−1

τ

τ

))=

ϕ(e(ukτ ))− ϕ(e(uk−1

τ )

τ+(1−

√τ

2

)ξ(e(uk

τ − uk−1τ

τ

))(12.157)


provided τ is sufficiently small, namely so small that ϕ(·) + ξ(·)/√4τ : Rn×n → Ris convex. Summing it for k = 1, ..., l yields the discrete mechanical-energy balance

�

2

∥∥∥ulτ−ul−1

τ

τ

∥∥∥2L2(Ω;Rn)

+ τ

l∑k=1

((1−

√τ

2

)ξ(e(uk

τ−uk−1τ

τ

))+ γ1

∣∣∣∇e(uk

τ−uk−1τ

τ

)∣∣∣2)+

∫Ω

ϕ(e(ulτ )) +

γ

2

∣∣∇e(ulτ )∣∣2 + τ

η|e(ul

τ )|η +τ

η|∇e(ul

τ )|η dx

≤ τl∑

k=1

(∫Ω

gkτ ·ukτ−uk−1

τ

τ− F

(e(uk

τ ), ϑkτ

):e(uk

τ−uk−1τ

τ

)dx

+

∫Γ

hkτ ·ukτ−uk−1

τ

τdS

)+

�

2

∥∥v0∥∥2L2(Ω;Rn)

+

∫Ω

ϕ(u0τ ) +γ

2

∣∣∇e(u0τ )∣∣2+ τ

η|e(u0τ )|η +

τ

η|∇e(u0τ )|η dx. (12.158)

Further we test the heat part (12.150b) by 1 and add it to (12.158). Observingcancellation of dissipative terms and also of the adiabatic F -terms, we arrive atthe discrete total-energy balance:

�

2

∥∥∥ulτ−ul−1

τ

τ

∥∥∥2L2(Ω;Rn)

+∥∥ϑl

τ

∥∥L1(Ω)

+

∫Ω

ϕ(e(ulτ )) +

γ

2

∣∣∇e(ulτ )∣∣2 + τ

η|e(ul

τ )|η +τ

η|∇e(ul

τ )|η dx

≤ τ

l∑k=1

(∫Ω

gkτ ·ukτ−uk−1

τ

τdx+

∫Γ

hkτ ·ukτ−uk−1

τ

τ+ fk

τ dS

)+

�

2

∥∥v0∥∥2L2(Ω;Rn)+∥∥Cv(e(u0τ ), θ0)

∥∥L1(Ω)

+

∫Ω

ϕ(u0τ ) +γ

2

∣∣∇e(u0τ )∣∣2+ τ

η|e(u0τ )|η +

τ

η|∇e(u0τ )|η dx. (12.159)

Then we employ the discrete Gronwall inequality. �Proposition 12.55 (Further estimates). Under the assumptions of Lemma12.54, it also holds that∥∥uτ

∥∥W 1,2(I;W 2,2(Ω;Rn))

≤ C, (12.160a)∥∥∥� ∂

∂t

[∂uτ

∂t

]i∥∥∥L2(I;W 2,2(Ω;Rn)∗)+Lη′(I;W 2,η(Ω;Rn)∗)

≤ C, (12.160b)∥∥∥� ∂

∂t

[∂uτ

∂t

]i − τdiv(|e(uτ )|η−2e(uτ )

)+ τdiv2

(|∇e(uτ )|η−2∇e(uτ ))∥∥∥

L2(I;W 2,2(Ω;Rn)∗)≤ C, (12.160c)

and the estimates (12.16b,c,f) for ϑτ and ϑτ .


Proof. The strategy (12.17) to get estimate (12.16b) for ∇ϑτ is to be modified byan additional term K1(e(uτ ), ϑτ )∇e(uτ )·∇χ(ϑτ ) with χ(ϑ) := 1 − 1/(1+ϑ)ε. Weproceed as follows:∫

Q

K1(e(uτ ), ϑτ )∇e(uτ )·∇χ(ϑτ ) dxdt = ε

∫Q

∇e(uτ ).:K1(e(uτ ), ϑτ )⊗∇ϑτ

(1 + ϑτ )1+εdxdt

≤ ε

∫Q

1

4δ|∇e(uτ )|2 + δ

|K1(e(uτ ), ϑτ )|21 + ϑτ

|∇ϑτ |2(1+ϑτ )1+ε

dxdt

≤ ε

∫Q

1

4δ|∇e(uτ )|2 + δCK1

|∇ϑτ |2(1+ϑτ )1+ε

dxdt (12.161)

with CK1from (12.152f). Using (12.156a) and choosing δ > 0 small, we can execute

the interpolation strategy (12.17). Instead of (12.25), we now have∣∣∣ ∫Q

F (e(uτ ), ϑτ ):e(∂uτ

∂t

)dxdt

∣∣∣ ≤ C∥∥F (e(uτ ), ϑτ )

∥∥2L2(Q)

+ δ∥∥∥e(∂uτ

∂t

)∥∥∥2L2(Q;Rn×n)

≤ Cδ + Cδ

∥∥e(uτ )∥∥2∗L2∗(Q;Rn×n)

+ Cδ

∥∥ϑτ

∥∥2/ωL2/ω(Q)

+ δ∥∥∥e(∂uτ

∂t

)∥∥∥2L2(Q;Rn×n)

where we used the restriction pF

< 2∗/2 in (12.152h). Note that, by (12.156a),we have ‖e(uτ)‖L2∗ (Q;Rn×n) ≤ T 1/2∗‖e(uτ )‖L∞(I;L2∗(Ω;Rn×n)) already estimated.Then we can finish the interpolation game as in (12.26)–(12.30). This yields both(12.16b,c) and (12.160a).

The dual estimates (12.16f) and (12.160b,c) can be obtained routinely; notethat (12.160c) relies on the restriction p

F< 2∗/2 ≤ 2∗ − 1 used in (12.152h) so

that F (e(uτ ), ϑτ ) is certainly bounded in L∞(I;L2∗′(Ω;Rn×n)). �

Proposition 12.56 (Convergence for τ → 0). Let (12.152) hold and η >max(4, 2p

F+2, 2/(1−q

F)), and let also

u0τ → u0 in W 2,2(Ω;Rn) and∥∥u0τ

∥∥W 2,η(Ω;Rn)

= o(τ−1/η

). (12.162)

Then there is a subsequence such that

uτ → u strongly in W 1,2(I;W 2,2(Ω;Rn)), (12.163a)

ϑτ → ϑ strongly in Lq(Q) with any q < (n+2)/n, (12.163b)

and any (u, ϑ) obtained by this way is a weak solution to the initial-boundary-valueproblem (12.148)–(12.149). In particular, (12.148)–(12.149) has a weak solution.

Proof. Choose weakly* converging subsequence {(uτ , ϑτ )}τ>0 in the topology ofthe estimates (12.156a)–(12.16b,c). By (12.160c), we can choose this subsequenceso that also

�∂

∂t

[∂uτ

∂t

]i− τdiv(|e(uτ )|η−2e(uτ )

)+ τdiv2

(|∇e(uτ )|η−2∇e(uτ ))⇀ ζ (12.164)


in L2(I;W 2,2(Ω;Rn)∗) for some ζ and, like in Exercise 8.85, we can show that

ζ = �∂2u∂t2 . By the interpolated Aubin-Lions’ Lemma 7.8, combining (12.156b),

(12.16b) and (12.16f), one obtains (12.163b).

Like in (12.32), we use the by-part summation (11.126) to obtain the identity∫Ω

�∂uτ

∂t(T )·vτ (T ) dx−

∫ T

τ

∫Ω

�∂uτ

∂t(· − τ)·∂vτ

∂tdxdt+

∫Q

(12ξ′e(∂uτ

∂t

)+ ϕ′(e(uτ )) + F (e(uτ ), ϑτ ) + τ

∣∣e(uτ )∣∣η−2

e(uτ )):e(vτ ) + hτ

.:∇e(vτ ) dxdt

=

∫Ω

v0·vτ (τ) dx +

∫Q

gτ ·vτ dxdt+∫Σ

hτ ·vτ dxdS (12.165)

with hτ = γ1∇e(∂uτ

∂t )+γ∇e(uτ)+τ |∇e(uτ )|η−2∇e(uτ ) and vτ and vτ as in (12.32).

For v smooth, by (12.156c), one has (12.33) and also∣∣∣∣ ∫Q

τ |∇e(uτ )|η−2∇e(uτ ).:∇e(v) dxdt

∣∣∣∣≤ τ

∥∥∇e(uτ )∥∥η−1

Lη(Q;Rn×n×n)

∥∥∇e(v)∥∥Lη(Q;Rn×n×n)

= O(τ1/η) → 0, (12.166)

and thus can see that the regularizing η-terms disappear in the limit for τ → 0.Altogether, we can pass to the limit in (12.165) directly (without using Minty’strick) to the weak formulation of the mechanical part (12.139).

Here it is again essential that �∂2u∂t2 in duality with ∂u

∂t and also that both

F (e(u), ϑ)∈Lq′ (Q;Rn×n) and ϕ′(e(u))∈Lq′ (Q;Rn×n) are in duality with e(∂u∂t ) ∈Lq(Q;Rn×n), so that, by substituting v = ∂u

∂t into (12.139) with taking (12.147)into account, we obtain the mechanical-energy equality∫

Ω

�

2

∣∣∣∂u∂t

(T )∣∣∣2+ ϕ

(e(u(T ))

)+

γ

2

∣∣∇e(u(T ))∣∣2 dx

+

∫Q

ξ(e(∂u∂t

))+ γ1

∣∣∣∇e(∂u∂t

)∣∣∣2dxdt = ∫Ω

�

2|v0|2+ ϕ(e(u0)) +

γ

2

∣∣∇e(u0)∣∣2dx

+

∫Q

g·∂u∂t

− F (e(u), ϑ):e(∂u∂t

)dxdt+

∫Σ

h·∂u∂t

dxdS (12.167)

with ϕ(e) = ψ(e, 0) as in (12.9). For the limit passage in the heat equation, weneed to prove the strong L2-convergence of e( ∂

∂tuτ ) and of ∇e( ∂∂tuτ ). We use∫

Q

ξ(e(∂u∂t

))+ γ1

∣∣∣∇e(∂u∂t

)∣∣∣2dxdt ≤ lim infτ→0

∫Q

ξ(e(∂uτ

∂t

))+ γ1

∣∣∣∇e(∂uτ

∂t

)∣∣∣2dxdt≤ lim sup

τ→0

∫Q

(1−

√τ

2

)ξ(e(∂uτ

∂t

))+ γ1

∣∣∣∇e(∂uτ

∂t

)∣∣∣2 dxdt


≤ lim supτ→0

∫Ω

�

2|v0|2 − �

2

∣∣∣∂uτ

∂t(T )

∣∣∣2 + ϕ(e(u0τ )) +γ

2

∣∣∇e(u0τ ))∣∣2

− ϕ(e(uτ (T ))

)− γ

2

∣∣∇e(uτ (T ))∣∣2 + τ

η

∣∣e(u0τ )∣∣η+ τ

η

∣∣∇e(u0τ )∣∣ηdx

+

∫Q

gτ ·∂uτ

∂t− F (e(uτ ), ϑτ ):e

(∂uτ

∂t

)dxdt+

∫Σ

hτ ·∂uτ

∂tdxdS

≤∫Ω

�

2|v0|2 + ϕ(e(u0)) +

γ

2

∣∣∇e(u0)∣∣2

− �

2

∣∣∣∂u∂t

(T )∣∣∣2 − ϕ

(e(u(T ))− γ

2

∣∣∇e(u(T ))∣∣2 dx

+

∫Q

g·∂u∂t

− F (e(u), ϑ):e(∂u∂t

)dxdt+

∫Σ

h·∂u∂t

dxdS

=

∫Q

ξ(e(∂u∂t

))+ γ1

∣∣∣∇e(∂uτ

∂t

)∣∣∣2 dxdt. (12.168)

Note that the 3rd inequality in (12.168) is due to (12.158) used for l = T/τ , whilethe 4rd inequality uses weak upper semicontinuity and (12.162), and eventually thelast equality in (12.168) is exactly (12.167). Thus we obtain equalities in (12.168)and, in particular, we obtain limτ→0

∫Q ξ(e(∂uτ

∂t )) dxdt =∫Q ξ(e(∂u∂t )) dxdt and

also limτ→0

∫Q|∇e(∂uτ

∂t )|2 dxdt = ∫Q|∇e(∂u∂t )|2 dxdt. By (12.152i), the quadratic

form ξ is coercive and thus we obtain both e(∂uτ

∂t ) → e(∂u∂t ) strongly in

L2(Q;Rn×n), and also ∇e(∂uτ

∂t ) → ∇e(∂u∂t ) strongly in L2(Q;Rn×n×n). Then thelimit passage in the semi-linear heat equation is simple.

In particular, using Lemma 12.53 and the above arguments, some weak solu-tion to (12.148)–(12.149) indeed exists because, due to the qualification (12.152k)of u0, the regularization u0τ with the properties (12.162) always exist. �

Remark 12.57. In fact, even a bigger growth of F (·, ϑ) than (12.152h) can beadmitted if further interpolation of this term would be done, cf. [369]. The exis-tence for large data (for a very similar model) has been investigated by Paw�lowand Zochowski [329] even allowing for γ1 = 0, and by Paw�low and Zajaczkowski[327] for a heat capacity depending on both θ and e(u) by using regularity un-der assumption of a smooth domain with zero Dirichlet boundary conditions. Theenhanced enthalpy transformation (12.143) is similar to [311] where dependenceon space/time variables, leading to additional terms in the transformed system,has been treated in an analogous way. Nonconvexity of ψ(·, θ) makes it possi-ble to model shape-memory alloys and thermodynamics of so-called martensitictransformation occuring typically in such materials, as already mentioned in Ex-ercise 11.43.

Remark 12.58 (Internal energy balance). Like (12.3), we have now ϑ = w −ϕ(e) − 1

2H∇e.:∇e, which suggests to subtract from the internal-energy balance

∂w∂t = ψ′e(e(u), θ):e

(∂u∂t

)+H∇e(u)

.:∇e

(∂u∂t

)+ ξ1(e(

∂u∂t ),∇e

(∂u∂t ))+div j the balance


of the stored-energy rate versus the power of conservative parts of (hyper)stresses,i.e. ∂

∂t (ϕ(e(u))+12H∇e(u)

.:∇e(u) = ϕ′(e(u)):e(∂u∂t )+H∇e(u)

.:∇e(∂u∂t ). In this way,

we obtain ∂ϑ∂t + div j = (σ−ϕ′(e(u))):e

(∂u∂t

)+ (h−H∇e(u))

.:∇e

(∂u∂t

)which, after

eliminating temperature as we did by the substitution (12.143), would again resultto the transformed heat equation (12.148b). This reveals the physical character ofthe transformed system (12.148) and a certain conceptual similarity with thermo-dynamics of fluid where internal energy is sometimes used instead of temperaturefor analysis, cf. e.g. [80, 82].

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Index

absolute continuity 12, 22accretivity 97, 114

heat equation in L1(Ω) 105Laplacean in W 1,q(Ω) 113monotone mappings in Lq(Ω) 101

adjoint operator 5advection 74, 80, 105, 108, 279, 324

non-potentiality 129Alaoglu-Bourbaki theorem 7Allen-Cahn equation 287almost all (a.a.) 11almost everywhere (a.e.) 11anti-periodic condition 291, 314Asplund theorem 5Aubin-Lions’ lemma 208

with interpolation 210Baiocchi transformation 164, 168Banach selection principle 7, 64Banach space 2

uniformly convex 3Banach-Steinhaus principle 4Banach theorem 7

about a fixed point 8Benard problem 178bidual 5Bochner

integrable 22measurable 22space 23

Bolzano-Weierstrass theorem 8bootstrap argument 26, 90boundary 15boundary conditions

Dirichlet 43, 56, 275mixed 43

Navier 181Neumann 43Newton 43, 56Signorini-type 158

boundary-value problem 43bounded mapping 5bounded set 2Brezis-Ekeland principle 296Brouwer fixed-point theorem 8Browder-Minty theorem 40Burgers equation 320

regularized 286by-part integration 21, 205by-part summation 385Cahn-Hilliard equation 287, 420Caratheodory mapping 19Caratheodory solution 90Carleman system 332Cauchy-Bunyakovskiı inequality 4Cauchy sequence 2Cauchy problem 213

for 2nd-order problems 326, 377chain rule 207, 303, 362, 370

discrete 231, 397Clarke gradient 137Clarkson theorem 5classical solution 43Clement quasi-interpolant 229

of the 1st order 235closed 2closure 2coercive 33, 115

weakly 115, 216semi- 216, 240

T. Roubí ek, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics 153, DOI 10.1007/978-3-0348-0513-1, © Springer Basel 2013

469

470 Index

compactembedding 9mapping 7relatively 7set 7weakly in L1 14

comparison principle 72competition-in-ecology model 186complementarity problem 137complete 2condition

anti-periodic 291, 314boundary 43, 158, 275initial 213Leray-Lions 55periodic 289, 314

cone 6conjugate exponent 12, 299conjugate function 294conservation law 108, 320

regularized 285, 325consistency 44

for the variational inequality 138continuous 3

absolutely 12, 22casting 166demi- 32embedding 9equi-absolutely 14hemi- 32Lipschitz 5mapping 3radially 32semi- 3totally 7, 32uniform 5upper semi- 8weakly 32, 61weakly lower semi- 4

contraction 5convergence in the measure 13convergent sequence 2

weakly* 4

convex function 6poly- 175semi- 441strictly 6

convex set 6uniformly 3

cooperation-in-ecology model 186Crandall-Liggett formula 316critical growth 53, 68critical point 115d-monotonicity 32

of p-Laplacean 75Darcy-Brinkman system 281decoupling 238, 364, 404, 409demicontinuous 32dense 2

embedding 9direct method 115, 135

for parabolic problems 297, 364for weakly continuous maps 65

Dirichlet boundary conditions 43for parabolic equation 275

directional derivative 5dissipative mapping 97distribution 10distributional derivative 15

of vector-valued function 201distributional solution 106, 110

selectivity 113divergence 21domain 15

of Ck-class 16drift-diffusion model 192, 412dual problem 145, 162dual space 3duality mapping 95, 126

for Lp(Ω) or W 1,2(Ω) 99potential of 134

duality pairing 3Dunford-Pettis theorem 14Duvaut transformation 348eddy-current approximation 427, 437Ehrling lemma 207elasticity 176

Index 471

elliptic 43energetic solution 391energy method 31enthalpy transformation 277, 319, 395

enhanced 439, 447steady-state problems 108, 185

epigraph 6equation

Allen-Cahn 287biharmonic 60, 283Burgers 286, 320Cahn-Hilliard 287, 420Darcy-Brinkman 281drift-diffusion 192, 412doubly nonlinear 326, 351, 367,

377elliptic 43Euler-Lagrange 121, 129, 176fully nonlinear 93, 300Hamilton-Jacobi 109, 321heat 68, 73, 105, 185, 277, 319,

324, 374hyperbolic 43, 327, 390, 388integro-differential 261Klein-Gordon 333, 388Lame (system) 177Landau-Lifschitz-Gilbert 374,432Lotka-Volterra (system) 186,408Navier-Stokes 184, 279Nernst-Planck-Poisson 416Oberbeck-Boussinesq 178, 405Oseen 179, 405parabolic 43parabolic/elliptic 374partial differential xiPenrose-Fife (system) 419phase field (system) 331, 416Poisson 193predator/prey 186, 408pseudoparabolic 287, 356, 390quasilinear xireaction-diffusion (system)186,408semiconductor (Roosbroeck sys-

tem) 192, 412

semilinear xithermistor (system) 188thermo-visco-elasticity 328, 393

Euclidean space 2Euler-Lagrange equation 121, 176

of higher order 129Faedo-Galerkin method 240Fatou theorem 13Fenchel inequality 295finite-element method 67, 128fixed point 8formula

by-part integration 21, 205by-part summation 385Crandall-Liggett 316Gear 235Green 21implicit Euler 215semi-implicit 228

Fourier law 189fractional step method 239free boundary 144free boundary problems 144Friedrichs inequality 22friction 365Fubini theorem 14fully nonlinear equation 93, 300Galerkin approximation 33

for elliptic equation 67for evolution equation 240for heat equation 70for inequalities 153, 159for thermo-coupled systems 403

Gagliardo-Nirenberg inequality 17generalized 253

Garding inequality 216Gateaux differential 5Gear formula 235Gelfand triple 204generalized standard material 329gradient 15Green formula 21Gronwall inequality 25

discrete 26

472 Index

Hahn-Banach theorem 6Hamilton-Jacobi equation 109, 321Hausdorff space 2heat equation 68, 73

degenerate 374evolutionary 277, 319, 374L1-theory 105, 319nonlinear test 324positivity of solution 325, 402uniqueness 74

Helly selection principle 222hemicontinuous 32hemivariational inequality 137Hilbert space 2homeomorphical embedding 3homeomorphism 3hyperbolic equation 43, 327, 388, 390implicit Euler formula 215

sequential splitting 239implicit variational inequalities 170indicator function 134inequality

Cauchy-Bunyakovskiı 4elliptic variational 137Fenchel 295Friedrichs 22Gagliardo-Nirenberg 17, 253Garding 216Gronwall 25implicit variational 170hemivariational 137Korn 22parabolic variational 343Poincare 21quasivariational 154variational 133Young 12

initial condition 213inner product 2integrable function 11

uniformly 14integral solution 308, 332interior 2interpolation 13, 17, 24, 210, 257, 267

Kakutani fixed-point theorem 8Kirchhoff transformation 73, 277, 319Klein-Gordon equation 333, 388Komura theorem 23Korn inequality 22Lagrangean 144, 162Lame system 177Landau-Lifschitz-Gilbert equation 374

modified 386Lax-Milgram theorem 40Lebesgue

integral 11measurable function 11measurable set 10measure 11outer measure 10point 22space 12

Legendre-Fenchel transformation 294Leray-Lions condition 55linear operator 3Lipschitz continuous 5Lipschitz domain 15locally convex space 2

Hausdorff 2Lotka-Volterra system 408

steady-state 186m-accretive 97, 114mapping

accretive 97compact 7continuous 3d-monotone 32demicontinuous 32dissipative 97duality 95, 126hemicontinuous 32Lipschitz continuous 5m-accretive 97, 114maximal accretive 98, 112maximal monotone 133monotone 31, 133Nemytskiı 19, 24pseudomonotone 32, 64, 170, 222

Index 473

radially continuous 32set-valued 8strictly monotone 31totally continuous 7, 32type (M) 93, 170type (S+) 93upper semicontinuous 8weakly continuous 32, 61

measure 10absolutely continuous 12Dirac 10Lebesgue 11on the right-hand side 55, 109regular Borel 10

mild solution 317, 333Milman-Pettis theorem 5Minty trick 37, 152, 233

for inequalities 143, 159mollifiers 203monotone 31, 133

E- 375in the main part 49maximal 133, 289strictly 31strongly 32uniformly 32

Mosco convergence 147Mosco transformation 153Moser trick 324multilevel formula 235Navier-Stokes equations 184

evolution 279Nemytskiı mappings 19

in Bochner spaces 24Nernst-Planck-Poisson system 416Newtonean fluids 184non-autonomous 236, 319non-expansive 5non-Newtonean fluids 178, 287normal cone 6, 134norm 1

in Lp-spaces 11in W 1,p-spaces 15semi- 1

normed linear space 1Oberbeck-Boussinesq model 178, 405Oseen equation 179, 405p-biharmonic operator 60, 130p-Laplacean 75, 127

anisotropic 129regularized 79, 128, 274

parabolic 43partial differential equations xipenalty method 140, 153Penrose-Fife system 419periodic condition 289, 314periodic problems 289, 297, 313Pettis theorem 22phase field system 331, 416pivot 205Poincare inequality 21Poisson equation 193polyconvexity 174potential 115

of anisotropic p-Laplacean 129of duality mapping 134of higher-order problems 129of p-Laplacean 127of parabolic problems 295, 299of system of equations 176

precompact set 7predator/prey model 186, 409predual 3projector 3proper 134pseudomonotone 32, 64, 170, 222, 247pseudoparabolic equation 356, 390

in magnetism 374, 434of the 4th order 287

quasiconvexity 175quasilinear xiquasivariational inequality 154

evolutionary 365Rademacher theorem 21radially continuous 32rank-one convexity 175rate-independent systems 391reflexive 5

474 Index

regular Borel measure 10absolutely continuous 12Dirac 10

regularity 85abstract evolution equation 230

regularization 336elliptic 349systems 396, 433variational inequality 140, 160, 344

Rellich theorem 16Ritz method 120, 126, 128, 158Robin boundary condition 43Rothe method 215

decoupling 238, 364, 404, 409for accretive mappings 305for doubly nonlinear problems

352, 368for second-order problems 384, 390for variational inequalities 339semi-implicit variant 228, 279,

372, 375, 409, 419scalar product 2Schauder fixed-point theorem 8

Tikhonov modification 65selectivity 44

for distributional solutions 113for evolution equations 214, 249for evolution inequalities 339for 4th-order equations 59for integral solutions 309for mild solutions 318for 2nd-order equations 45for variational inequalities 138

semi-coercive 216, 240semi-convex 238, 441semicontinuous function 3

weakly 4semigroup 315

non-expansive 315of the type λ 315

semi-implicit formula 228for doubly-nonlinear problem372for doubly-nonlinear system365for heat equation 279

for parabolic equation 288for parabolic system239for phase-field system 419for predator-prey system 409for thermo-coupled system 404

semi-inner product 97semilinear xi, 62set-valued mapping 8shape-memory alloys 389, 447Signorini problem 158simple function 11, 22singular perturbations 84, 130, 283small strain 177smooth 5, 9Sobolev-Slobodeckiı space 18Sobolev space 15solution

Caratheodory 90classical 43distributional 106, 110energetic 391integral 308, 332mild 317, 333strong 214, 303, 335, 377very weak 106, 169, 252, 278weak 45, 214, 252, 339

spaceBanach 2bidual 5Bochner 23dual 3Hilbert 2Lebesgue 12locally convex 2normed linear 1predual 3reflexive 5Sobolev 15Sobolev-Slobodeckiı 18strictly convex 3uniformly convex 3

Stefan condition 167Stefan problem 320

one-phase 167, 348

Index 475

steepest-descent method 120Stefanelli variational principle 364strain tensor 176strictly monotone 31strong convergence 4

by d-monotonicity 41of Ritz’ method 129

strong solutionof 1st-order equations 214of 2nd-order equations 377of accretive equations 303of variational inequalities 335

strongly monotone 32subdifferential 134super-critical growth 68, 72surface integral 21sweeping process 391symmetry condition 121

of the 2th-order system 175of the 4th-order problem 130

tangent cone 6theorem

Alaoglu-Bourbaki 7Asplund 5Aubin-Lions’ (lemma) 208Banach fixed point 8Banach selection principle 7Banach-Steinhaus (principle) 4Bolzano-Weierstrass 8Brezis 33Brouwer fixed-point 8Browder-Minty 40Clarkson 5Dunford-Pettis 14Ehrling (lemma) 207Fatou 13Fubini 14Green 21Hahn-Banach 6Kakutani fixed-point 8Komura 23Lax-Milgram 40Leray-Lions 54Lumer-Phillips 317

Milman-Pettis 5Minty (trick) 37Papageorgiou (lemma) 222, 242Pettis 22Rademacher 21Rellich-Kondrachov 16Sobolev embedding 16Schauder fixed-point 8Tikhonov fixed-point 65Vitali 14

thermo-visco-elasticity 393linearized 328fully nonlinear 438

totally continuous mapping 7, 32trace operator 17

on Sobolev-Slobodeckiı space 254transformation

Baiocchi 164, 168Duvaut 348enthalpy 108, 277, 319, 395, 439Kirchhoff 73, 277, 319Legendre-Fenchel 294Mosco 153

transposition method 109transversality 137uniformly continuous 5uniformly convex space 3

Hilbert space 64Lp(I;V ) 23Lp(Ω;Rm) 12

uniformly monotone 32unilateral problem 137unit outward normal 20upper semicontinuous mapping 8variational inequality 133

boundary 357hemi- 137implicit 170of type II 357quasi- 154

variational methods 115very weak solution 106, 169

of heat equation 278of parabolic equation 252

476 Index

Vitali theorem 14weak convergence 4weak derivative 303weak formulation 44

of parabolic equation 252weak solution 45, 214, 339

of parabolic equation 252weakly continuous mappings 32, 61weakly lower semicontinuous 4Young inequality 12Yosida approximation 150

of a functional 147modification of 83

nonlinear partial differential equations with applications ||

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